Notices respecting Neic Books. 149 



tion of their properties. This part of the subject does not present 

 any serious difficulty, especially as in it we have to deal with 

 functions of but one independent variable. It is easily shown that 

 P„ satisfies the equation 



^{ (W) S} + " ( " +l ) p "= 0: $ 



this equation will have a second solution which can be obtained in 

 a finite form, and which Mr. Todhunter calls " a coefficient of the 

 second kind." It is easily shown that any rational function of x 

 can be expressed in terms of Legendre's coefficients. This suggests 

 the question, Can any function of x be so expressed ? The dis- 

 cussion of this question yields the following answer : — If 



/(ff)=a +« 1 P 1 + « a P a + , 



the constants a , a v « 2 , . . . have determinate values. Next, no 

 second form, such as & Q -f& 1 P 1 + & 2 P 2 4- • • • can exist. In the 

 third place, the expression is always practicable, provided the 

 series, whose general term is 



f, 



^±i ? n (x) \ V n (x)f(x)dx 



is finite ; but whether the series is necessarily finite or not is not 

 shown ; and here the discussion comes to an end with this difficulty 

 left in it. Further on, however (art. 220), the series is shown to 

 be finite, provided x is less than unity. 



It is not hard to see why these functions should be thus minutely 

 treated. In the first place, they are important for their own sake ; 

 in the next, their discussion forms the best possible introduction 

 to that of Laplace's functions, of which they are a sort of parti- 

 cular case. If we suppose the position of the points to be expressed 

 by means of their polar coordinates (r, 0, <p) and (/, 0', <f>), the 

 quantity denoted by x in the above expression for TJ becomes 

 sin0 sin0' cos (^— 0') + cos0 cos0 f , 



and the coefficients of the expansion are no longer functions of a 

 single variable x, but of two, viz. and 0, or it may be 0' and 0', 

 and are, in fact, Laplace's coefficients. As is well known, the nth 

 of them (say T B ) satisfies the equation 



where /* stands for cos0. The above equation (1) is strictly analo- 

 gous to this equation, and is, indeed, a particular case of it. In 

 this case we shall have numerous solutions, any one being a " La- 

 place's Function of the nth order," while the particular case which 

 gives a coefficient in the expansion of U is a Laplace's Coefficient 

 of the nth order. 



Prom this account of the matter it will be seen how the author 

 gradually leads the student up to the difficulties of the subject ; 



