In the solution of a great many types of problems in applied mathematics we are led to the solution of linear differential equations or sets of linear differential equations. Usually these equations are equations having constant coefficients, and in a majority of these cases we are led to solutions of the exponential type, which includes trigonometric and hyperbolic functions as special cases. This is the situation that arises when we study the oscillations of linear electrical or mechanical systems.
However, this is not always the case, and equations of other forms are encountered; because of the frequency of their appearance, their solutions have been classified and tabulated for future reference. In this appendix we will present some of the properties of these functions in view of their practical importance. The functions to be considered are Bessel, Legendre, gamma, beta, and error functions.
As a starting point of the discussion, let us consider the linear differential equation
where n is a constant.
This equation is known in the literature as Bessel’s differential equation. Since it is a linear differential equation of the second order, it must have two linearly independent solutions. The standard form of the general solution of (2.1) is
where C1 and C2 are arbitrary constants and the function Jn(x) is called the Bessel function of order n of the first kind and Yn(x) is the Bessel function of order n of the second kind. These functions have been tabulated and behave somewhat like trigonometric functions of damped amplitude. To see this qualitatively, let us transform the dependent variable by the substitution
This transformation converts (2.1) into
In the special case in which
this becomes
Hence
and
where C1 and C2 are arbitrary constants. Also we see that as x → ∞ in (2.4), and n is finite, we would expect the solution of (2.1) to behave qualitatively as (2.8) to a first approximation.
If we introduce the operator
then Bessel’s differential equation (2.1) may be written in the form
In order to solve this equation, let us assume an infinite-series solution in the form
Now
Hence, on substituting (3.3) into (3.2), we have
If we now equate the coefficients of the various powers of x, xr, xr+1, xr+2, etc., to zero in (3.6), we obtain the set of equations
This is valid for s = 0, 1, 2, . . . in view of the fact that
since the leading coefficient in the expansion (3.3) is C0.
Letting s = 0 in (3.7), we obtain
This equation is known as the indicial equation, and since
it follows that
and from the equation
The relation between Cs and Cs–2 now shows, taking s = 3, 5, . . . in succession, that all coefficients of odd rank vanish.
Taking first of all r = n, we may write (3.7) in the form
From (3.14) we see that the coefficients C2, C4, C6, etc., are all determined in terms of C0. Inserting these values of the coefficient into the assumed form of solution (3.3), we obtain the solution
The coefficients are finite except when n is a negative integer. Excluding this case, we standardize the solution by taking
in general and
when n is a positive integer. Inserting this value of C0 into (3.15) and generalizing the factorial numbers when n is not an integer by writing
we obtain
This series converges for any finite value of x and represents a function Jn(x) of x that is known as the Bessel function of the first kind of order n. When n is not an integer, the second solution may be obtained by replacing n by -n in accordance with (3.11). It is therefore
The leading terms of Jn(x) and J–n(x) are, respectively, finite (nonzero) multiples of xn and x–n; the two functions are not mere multiples of each other, and hence the general solution of the Bessel differential equation may be expressed in the form
where A and B are arbitrary constants provided that n is not an integer.
However, when n is an integer, and since n appears in the differential equation only as n2 there is no loss of generality in taking it to be a positive integer, J–n(x) is not distinct from Jn(x) In this case the denominators of the first n terms of the series for J–n(x) contain the factors
for s = 0, 1, 2, . . . , n – 1. Hence these terms vanish. Therefore
If we now let
then
for n = 0, 1, 2, 3, . . . . In this case we no longer have two linearly independent solutions of the differential equation, and an independent second solution must be found.
In the preceding section we have seen that if n is not an integer, a general solution of the Bessel differential equation of order n is given by (3.21). If, however, n is an integer, then in view of (3.25) we have
where C is an arbitrary constant. We therefore do not have the general solution of Bessel’s differential equation, since such a solution must consist of two linearly independent functions multiplied by arbitrary constants. Consider the function
Now if n is not an integer, the function Yn(x) is dependent on Jn(x), and since it is a linear combination of Jn(x) and J–n(x), it is a solution of Bessel’s differential equation of order n. If now n is an integer, because of the relation (3.25), we have
Hence, when n is an integer, we define Yn(x) to be
With this definition of Yn(x) we have, on carrying out the limiting process,
where γ is Euler’s constant defined by
Also, when n is any positive integer, we have
where, for r = 0, instead of
we write
The presence of the logarithmic term in the function Yn(x) shows that these functions are infinite at x = 0. The general solution of Bessel’s differential equation may now be written in the form
where C1 and C2 are arbitrary constants.
In Sec. 2 we saw that the transformation
transformed Bessel’s differential equation into the form
We would suspect qualitatively that for large values of x the Bessel functions would behave as the solutions of the equation obtained from (5.2) by neglecting the 1/x2 term, that is, as solutions of the equation
and hence as
More precise analysis shows that
That is, for large values of the argument x, the Bessel functions behave like trigonometric functions of decreasing amplitude.
From the series expansions of the functions Jn(x) and Yn(x) we also have the following behavior for small values of x:
The value of Yn(x) is always infinite at x = 0. For small values of x this function is of the order l/xn if n ≠ 0 and of the order log x if n = 0.
Some important recurrence relations involving the function Jn(x) may be obtained directly from the series expansion for the function. From (3.19) we have
If we write
we have
If in the last summation we place
we obtain
In the same manner, we can prove that
If we add (6.5) to (6.6), we have
If we place n = 0 and use Eq. (3.25), we have
If we multiply (6.5) by x–n–1 we obtain
Hence
If we subtract (6.6) from (6.5), we obtain
Many other recurrence formulas may be obtained.
The case when n is half an odd integer is of importance because these particular Bessel functions can be expressed in finite form by elementary functions.
If we place 
the general series for Jn(x) given by (3.19), we obtain
Now since
and
we have
However we have
and
Hence from (7.4) we have
If we place 
in the general series for Jn(x), we may also show that
If in the recurrence formula (6.12) we place 
we obtain
Hence
If in (6.12) we let 
we obtain
or
In the same way we may show that
In some physical investigations we encounter complex combinations of Bessel functions of the first and second kinds so frequently that it has been found convenient to tabulate these combinations and thus define new functions.
These new functions are defined by the equations
and are called Bessel functions of order n of the third kind, or Hankel functions of order n. These functions are complex quantities.
In applied mathematics differential equations whose solutions are expressible in terms of Bessel functions are frequently encountered. Lists of these equations have been compiled.† Several of these differential equations will be listed here for reference.
In order to shorten the writing, the notation
where C1 and C2 are arbitrary constants, will be introduced. In terms of this notation the following differential equations have the indicated solutions:
Equation (9.4) is known in the mathematical literature as Stokes’s equation and is of considerable importance in mathematical physics because of its occurrence in the theory of diffraction and refraction of waves as well as in the theory of transmission across potential barriers.‡
Let us consider the differential equation
This equation is of the form (9.3) with
Hence Jn(jx) is a solution of this equation. The function
is taken as the standard form for one of the fundamental solutions of (10.1). The function In(x) defined in this manner is a real function and is known as the modified Bessel function of the first kind of order n. Another fundamental solution of Eq. (10.1) is known as the modified Bessel function of the second kind and is defined by
The general solution of Eq. (10.1) may be written in the form
where A and B are arbitrary constants.
In contrast to the Bessel function Jn(x) and Yn(x) the functions In(x) and Kn(x) are not of the oscillating type, but their behavior is similar to that of the exponential functions. For large values of x we have
For small values of x we have
In determining the distribution of alternating currents in wires of circular cross section, the following differential equation is encountered:
This equation is a special case of Eq. (9.3) with
and
Hence
The general solution of Eq. (11.1) may therefore be written in the form
where A and B are arbitrary constants. The functions 
are complex functions. Decomposing them into their real and imaginary parts, we obtain
and
These equations define the functions ber(x), bei(x) and ker(x), kei(x).
In Appendix C it is pointed out that the expansion of an arbitrary function into a Fourier series is only a special case of the expansion of an arbitrary function in a series of orthogonal functions under certain restrictions. It will now be shown that it is possible to expand an arbitrary function in a series of Bessel functions. If in Eq. (2.4) we place ax instead of x, we obtain
This equation has one solution,
In the same manner
satisfies the equation
If we multiply (12.1) by v and (12.4) by u and subtract the second product from the first, we obtain
Let us now integrate both members of (12.5) with respect to x from 0 to x. We thus obtain
However, we have
That is,
If we now differentiate the last equation with respect to b and then set
we obtain
From (12.9) we have
Now the second member vanishes if a and b are roots of the equation
That is, if a and b are distinct positive zeros of Jn(α) we have
and since
we have
We are now in a position to expand an arbitrary function F(x) in the interval from x = 0 to x = 1 in a series of the form
where the αs are the successive positive roots of (12.13). To obtain the general coefficient Ck of this expansion, we multiply both members of (12.17) by xJn(αkx)dx and integrate from x = 0 to x = 1. We have by virtue of (12.16)
The last integral, which is independent of x, may be evaluated by means of (12.11), (12.3), and (6.9). Its value is
Hence the typical coefficient of the series expansion (12.17) is given by
This expansion is analogous to the expansion of an arbitrary function in a Fourier series.
The functions Jn(x) of integral order are sometimes called the Bessel coefficients. These functions occur as coefficients in the expansion of the following function in powers of t:
This function of t can be expressed as a series of the form
This is an infinite series of ascending and descending powers of t and is known in the mathematical literature as a Laurent series (see Chap. 1, Sec. 8). In order to obtain the coefficients An, we take the product of the following two expansions:
The constant term A0 of the expansion (13.2) is
The term An is
Similarly the coefficient A–n is
Accordingly we have
If we place t = ejθ, j = (–1)½ in the result (13.7), we obtain the following series:
Since ejxsinθ = cos (xsinθ) +jsin (xsinθ) if the real and imaginary parts of (13.8) are separated, the results are
and
If in (13.9) and (13.10) θ is replaced by its complement, π/2 – θ, the following equations are obtained:
These series are usually called Jacobi series in the mathematical literature. Each of these series may be regarded as a Fourier series. If we multiply all terms of (13.9) by cos nθ dθ and all terms of (13.10) by sin nθ dθ and integrate between 0 and π, the following results are obtained as a consequence of the orthogonality relations of the sines and cosines:
and
From this it can be deduced that
whether n is an odd or even integer. Therefore
These integrals are originally due to Bessel. The integral (13.16) occurred in an astronomy problem involving what is known as the eccentric anomaly.
In the last sections we discussed the solutions of Bessel’s differential equation or Bessel functions. Another differential equation that arises very frequently in various branches of applied mathematics is Legendre’s differential equation. This equation occurs in the process of obtaining solutions of Laplace’s equation in spherical coordinates and hence is of great importance in mathematical applications to physics and engineering. The following sections are devoted to the study of the solutions of Legendre’s differential equation and to a discussion of their most important properties.
The differential equation
is known in the literature as Legendre’s differential equation of degree n. We shall consider here only the important special case in which the parameter n is zero or a positive integer. As in the case of Bessel’s differential equation let us assume an infinite-series solution of this differential equation in the form
For (14.2) to be a solution of (14.1), it is necessary that when (14.2) is substituted into (14.1) the coefficient of every power of x vanish. Equating the coefficient of the power xm+r–2 to zero, we obtain
Since the leading coefficient in the series (14.2) is a0, we have
in (14.3).
With this stipulation, placing r = 0 in (14.3), we have
Placing r = 1 in (14.3), we obtain
Equation (14.5) gives m = 0 or m = 1, with a0 arbitrary in any case. Let us take m = 0; then ax is arbitrary. Placing this value of m in (14.3), we have
This enables us to determine any coefficient from the one which precedes it by two terms. We therefore have
It may be shown by the ratio test that each of these series converges in the interval (–1,+1). Had we taken the possibility m = –1 in (14.6), we would not have obtained anything new but only the second series in (14.8).
Since a0 and a1 are arbitrary, this is the general solution of Legendre’s equation. We notice that the first series reduces to a polynomial when n is an even integer and the second series reduces to a polynomial when n is an odd integer. Now if we give the arbitrary coefficients a0 or a1 as the case may be, such a numerical value that the polynomial becomes equal to unity when x is unity, we obtain the following system of polynomials :
These are called Legendre polynomials. Each satisfies a Legendre differential equation in which n has the value indicated by the subscript.
The general polynomial Pn(x) is given by the series
where N = n/2 for n even and N = (n– l)/2 for n odd.
It is thus seen that the Legendre polynomial Pn(x) is even or odd according as its degree n is even or odd. Since
we therefore conclude that
An important formula for Pn(x) may be deduced directly from Legendre’s differential equation. Let
Then
Hence
If we differentiate (15.3) with respect to x, we obtain
If we now differentiate this equation r times in succession, we have
where
In particular, if r = n, (15.5) reduces to
This is Legendre’s equation (14.1). Hence vn satisfies Legendre’s equation. But since vn is
vn is a polynomial of degree n, and since Legendre’s equation has one and only one distinct solution of that form, Pn(x), it follows that Pn(x) is a constant multiple of vn. Hence we have
To determine the constant C we merely consider the highest power of x on each side of the equation, that is,
Substituting this value of C into (15.9), we obtain
This is Rodrigues’ formula for the Legendre polynomials.
The general solution of Legendre’s equation may be written in the form
where A and B are arbitrary constants and Qn(x) is called Legendre’s function of the second kind. This function is obtained by methods that are beyond the scope of this discussion. It is defined by the following series when |x| < 1:
if n is even and
if n is odd, where
If, however, |x| > 1, the above series do not converge. In this case the following series in descending powers of x is taken as the definition of
Both Pn(x) and Qn(x) are special cases of a function known as the hypergeometric function. The function Pn(x) is the more important and occurs more frequently in the literature of applied mathematics.
The Legendre polynomial Pn(x) is the coefficient of Zn in the expansion of
in ascending powers of Z. This may be verified for the lower powers of n by expanding (17.1) by the binomial theorem. To prove it for the general term, we write
Now it is obvious from the nature of the binomial expansion that An is a polynomial in x of degree n. Also, if we place x = 1 in (17.1), we obtain
Hence An is equal to 1, when x = 1. Now, if we can show that An satisfies Legendre’s equation, it will be identical with Pn(x) as the An’s are the only polynomials of degree n that satisfy the equation and have the value 1 when x = 1 From (17.1) we obtain by differentiation
and
If we now substitute from (17.2) into (17.4) and equate the coefficients of Zn–1 on both sides of the equation, we obtain
Substituting into (17.5) from (17.2) and equating the coefficients of the power Zn–1 on both sides, we obtain
If in (17.7) we replace n by n + 1, we obtain
Now if we differentiate (17.6) with respect to x and eliminate dAn–2/dx by (17.7), we have
We now multiply (17.8) by – x and add it to (17.9) and obtain
Differentiating (17.10) with respect to x and simplifying the result by means of (17.8), we finally obtain
This shows that An is a solution of Legendre’s equation. Hence, for the reasons stated above, it is the same as Pn(x). We therefore have
The above formulas for the An’s are therefore valid for Pn(x) and give important relations connecting Legendre polynomials of different orders. From (17.1) and (17.2) we have the important relation
This equation is valid in the ranges
because of the region of convergence of the binomial expansion (17.2). The function ϕ is called the generating function for Pn(x). This result is of great importance in potential theory.
If we let
and substitute this into (17.13), we have
Now we have
By the binomial theorem we obtain
and
Multiplying (18.4) and (18.5) and picking out the coefficient of Zn, we have
Every coefficient is positive so that Pn is numerically greatest when each cosine is equal to unity, that is, when θ = 0. But since
it follows that
The first few functions Pn(cos θ) are
Like the trigonometric functions cos mx and sin mx, the Legendre polynomials Pn(x) are orthogonal functions. Because of this property it is possible to expand an arbitrary function in a series of Legendre polynomials.
We shall now establish the orthogonality property
To do this, recall that Pn(x) satisfies the Legendre differential equation (14.1). This equation may be written in the form
If we now multiply this by Pm(x) and integrate between the limits –1 and +1, we obtain
Now we may integrate the first term by parts in the form
The first term of (19.4) vanishes at both limits because of the factor 1 – x2; hence (19.3) reduces to
If in (19.5) we interchange n and m, we obtain
Subtracting (19.6) from (19.5), we get
This establishes (19.1).
If n = m, Eq. (19.1) fails to hold. We shall now show that
To do this, we square both sides of (17.13) and obtain
We now integrate both sides of this equation with respect to x over the interval (–1, 1) and observe that the product terms on the right vanish in view of the orthogonality property (19.1). We thus obtain
if |Z | < 1. But the integral on the left has the value
Equating the coefficient of the power Z2n on both sides of (19.11), we have
If F(x) is sectionally continuous in the interval (–1, 1) and if its derivative F(x) is sectionally continuous in every interval interior to (–1, 1), it may be shown that F(x) may be expanded in a series of the form
To obtain the general coefficient am, we multiply both sides of (20.1) by Pm(x) and integrate over the interval (–1, 1). We then obtain
in view of (19.1) and (19.8). The general coefficient of the expansion (20.1) is given by
The expansion (20.1) is similar to an expansion of an arbitrary function into a Fourier series.
In the solution of some potential problems it is convenient to use certain polynomials closely related to the Legendre polynomials. We shall discuss them briefly in this section.
If we differentiate Legendre’s equation
m times with respect to x and write
we obtain
Since Pn is a solution of Legendre’s equation (21.1), this equation is satisfied by
If now in (21.3) we let
we obtain
This equation differs from Legendre’s equation in an added term involving m. It is called the associated Legendre equation. By Eq. (21.5) we see that it is satisfied by
This value of w is the associated Legendre polynomial, and it is denoted by 
. We therefore have
We notice that, if m > n, we have
The gamma function Γ(n) has been defined by Euler to be the definite integral
This definite integral converges when n is positive and therefore defines a function of n for positive values of n. By direct integration it is evident that
By an integration by parts the following identity may be established :
Comparing the result with (22.1), we have
This is the fundamental recursion relation satisfied by the gamma function. From this relation it is evident that if the value of Γ(n) is known for n between any two successive positive integers, the value of Γ(n) for any positive value of n may be found by successive applications of (22.4). Equation (22.4) may be used to define Γ(n) for values of n for which the definition (22.1) fails. We may write (22.4) in the form
Then if
formula (22.5) gives us Γ(n) since n + 1 is positive. We may then find Γ(n) where – 2 < n < – 1 since now n + 1 on the right-hand side of (22.5) is known, and so on indefinitely. We then have in (22.1) and (22.5) the complete definition of Γ(n) for all values of n except n = 0, – 1, –2,....
From Eq. (22.2) we have
Now, by the use of (22.4), we obtain
provided n is a positive integer. From this it is convenient to define 0! in the form
Gauss’s pi function is defined in terms of the gamma function by the equation
We thus see that if n is a positive integer
If we place n = 0 in Eq. (22.5), we have
By repeated application of (22.5) it is seen that the gamma function becomes infinite when n is zero or a negative integer.
GRAPH OF THE GAMMA FUNCTIONIf in the fundamental integral (22.1) we make the substitution
we obtain
If now 
we have
By making use of Appendix F, Eq. (11.22), we obtain
From the result (22.5) we obtain
etc. Figure 24.1 represents the graph of Γ(n).
Fig. 24.1
The beta function β(m,n) is defined by the definite integral
This integral converges and thus defines a function of m and n provided that m and n are positive.
If we let
In(25.1), we obtain
If in (25.1) we let x = sin2ϕ, we obtain
The substitution x = y/a in (25.1) gives
If x = y/(1+y) in (25.1)we obtain
These are the more common forms of the integral definition of the beta function.
Consider the gamma function as given by (24.2),
We may also write
and hence
If we now consider this integral as a surface integral in the first quadrant of the xy plane and introduce the polar coordinates
and introduce the surface element ds in the form
then (26.3) becomes
Now from (25.4) we have
and from (26.2) we have
Hence (26.6) may be written in the form
This formula is very useful for the evaluation of certain classes of definite integrals. For example, from (26.7) and (26.10) we obtain
If in (26.11) we let
we obtain
In a similar manner we obtain
In a similar manner many other integrals may be evaluated in terms of the gamma functions. If a table of gamma functions is available, then the computation of these integrals is considerably simplified.
Substituting (25.6) into the relation (26.10), we obtain
If we now let
in (27.1), we obtain
Now in Chap. 1 it is shown that
Hence, since
we have from (27.3) the important relation
Another very important function that occurs frequently in various branches of applied mathematics is the error function, erf (x), or the probability integral defined by
This integral occupies a central position in the theory of probability and arises in the solution of certain partial differential equations of physical interest.
From the definition of erf (x) we have
1. Prove that
2. Show that the differential equation
is satisfied by 
3. Show that
4. Show that
for integral values of n.
5. By multiplying the expansions for ex(t–t–1)/2 and e–x(t–t–1)/2 show that
6. Show that
7. Obtain the recurrence formulas of Sec. 6 by differentiating the relation
with respect to x, or to t, and comparing the coefficients of corresponding powers of t.
8. By modifying the variables in the above relation prove that
Replace t by t–1 and deduce that In(x) = I–n(x)
9. Prove that
10. Show that
11. Show that ex cos θ = I0(x)+ 2I1(x)cos θ + 2I2 (x)cos θ…
12. From the Jacobi series deduce that
13. A simple pendulum is performing small oscillations about the vertical while the length of the pendulum is increasing at a constant rate. Obtain the equation of motion of this pendulum, and express the solution in terms of Bessel functions.
14. A variable mass m(t) is attached to a spring of constant spring constant k. One end of the spring is fixed, and the mass is performing oscillations on a smooth horizontal plane. Discuss the oscillations of the mass if its magnitude is m = (a + bt)–1 where a and b are positive constants and is the time.
15 A nonlinear electrical circuit has a potential E cos wt impressed upon it. The current produced by this potential can be expressed in the form i = AebE cos wt, where A and b are positive constants. Prove that the mean value of the current is AI0(bE) and that the root-mean-square current is 
I0 is the modified Bessel function of zeroth order.
16. Prove that
17. Find the complete solution of the differential equation 
18. Show that x/2 = J1(x) + 3J3 (x) + 5J5 (x) + . . . .
19. Show that x sin x/2 = 22 J2 (x) – 42 J4 (x) + 62 J6(x) –. . . .
20. Show that x cos x/2 = 12 J(x) – 32J3(x) + 52J5(x)– . . .
21. Find the general solution of the equation d2 y/dx2 + (1 /x) dy/dx – k2y = a, where k and a are constants.
22. Show that
23. Establish the orthogonality property of the Legendre polynomials (19.1) by using Rodrigues’ formula for Pn(x) and successive integration by parts.
24. Show that
25. Show that
26. Prove that
27. Using Rodrigues’ formula, integrate by parts to show that
28. Show that, if Rm(x) is a polynomial of degree m less than n, we have
29. Show that
where k is a positive integer.
30. Show that
if n is an even positive integer.
31. show that
if n is an odd positive integer.
32. Show that
33. Evaluate the following definite integrals :
34. Show that
35. Show that
36. Show that by a suitable change in variable we have
37. Evaluate the integral
by expanding the integral in series, and show that
where R < x11/1,320.
38. Show by integrating by parts that
Show how this expression may be used to compute the value of erf (x) for large values of x.
1893. Byerly, W. E.: “Fourier’s Series and Spherical Harmonics,” Ginn and Company, Boston.
1911. Wilson, E. B.: “Advanced Calculus,” Chap. 14, Ginn and Company, Boston.
1922. Watson, G. N.: “Theory of Bessel Functions,” Cambridge University Press, New York.
1926. Woods, F. S.: “Advanced Calculus,” Chap. 7, Ginn and Company, Boston.
1927 Jeans, J. H.: “The Mathematical Theory of Electricity and Magnetism,” Cambridge University Press, New York.
1927. Whittaker, E. T., and G. N. Watson: “A Course in Modern Analysis,” 4th ed., Cambridge University Press, New York.
1931. Gray, A., G. B. Mathews, and T. M. Macrobert: “A Treatise on Bessel Functions,” The Macmillan Company, New York.
1934. McLachlan, N. W.: “Bessel Functions for Engineers,” Oxford University Press, New York.
1939. Smythe, W. R.: “Static and Dynamic Electricity,” McGraw-Hill Book Company, New York.
1941. Churchill, R. V.: “Fourier Series and Boundary Value Problems,” McGraw-Hill Book Company, New York.
1946. Relton, F. E.: “Applied Bessel Functions,” Blackie & Son, Ltd., Glasgow.
1953. Bickley, W. G.: “Bessel Functions and Formulae,” Cambridge University Press, New York.
† See E. Jahnke and F. Emde, “Tables of Functions,” pp. 146–147, Dover Publications, Inc., New York, 1943; W. G. Bickley, “Bessel Functions and Formulae,” Cambridge University Press, New York, 1953.
† See “Tables of Modified Functions of Order One-third and Their Derivatives,” Harvard University Press, Cambridge, Mass., 1945.
