549 



or dividing out by q(ni) cos a, (for 0(w) = cosa would only lead to 



*}. . . . (10) 

 Integrating this equation, determining the arbitrary constant by 

 the condition that <f>(m) = Q when m=0, and writing /3 for sin a . 0'(0), 

 we have 



Substituting in (8) and reducing, we find 



2 (12) 



sin 



But (8) was derived, not from (7) directly, but from (7) squared; 

 and on extracting the square root of both sides of (12), we must 

 choose that sign which shall satisfy (7), and therefore we must take 

 the sign -f-> as we see at once on putting m=n = Q. The equation 

 (12) on taking the proper root and (11) may be put under the form 



_._ 

 sin (wz/3) sin a sin (a-j- m(3) 



and to determine the arbitrary constants a, j3 we have, putting m= 1, 

 and m=r, \m = t 



_ _ 



sin/3 sin a sin(a+/3) 



We readily get from equations (13), 



(14) 





sn a 



whence the equations (6), (7) are easily verified. This verification 

 seems necessary in logical strictness, because we have no right to 

 assume a priori that it is possible to satisfy (6) and (7) for general 

 values of the variables ; and in deriving the equation ( 1 0), the equations 

 (6) and (7) were only assumed to hold good for general values of 

 m and infinitely small values of n. 



The equations (13), (14) give the following ^^-geometrical 

 construction for solving the problem: Construct a triangle of which 



