1824.] Mr. Herapath on the Solution o/'vj/" x — x. 423 



the solution of (27) free from the symmetrical form, and therefore 

 possessing many advantages over it in point of practical conve- 

 nience ; besides being complete for all values of r and n, which 

 no other solutions, hitherto given of (27), are. 



The solutions analogous to those usually given for a more 

 limited form of (27), are 

 4, x — f x + <? cc r x + <p a* r x + <p a.'"- r x} (32) 



and 

 4, x = XT & "' *i * Q * x, . . . . «' r ~ r x}*- (33) 



where q is the prime numerator of -, and <p, x^ perfectly arbitrary 



functions, the latter being limited to symmetrical forms. These 

 solutions are therefore confined to rational values of r and n, or 



at least of the quotient - ; and cannot hence be called complete. 



Their multinomial forms too render them much inferior to (31), 

 especially when they involve a function to be determined. 



Our (31), it will be seen, demonstrates the practicability of 

 applying Laplace's method to periodic functions, which has been 

 much questioned. On this method I have some observations to 

 make, which, for the present, brevity obliges me to withhold. 



§5. From what we have said in deducing (25), it appears 



that sin -, cos , and 1", the expressions usually given 



for the integral of 



are only particular cases of the general integral/* (1") or <p sin 



2 k >, t 



n 



Even 



1 11 in 



2 k \ z „ U'U . 2/cAs , Ik h z 



X 



t . a KK Z . 1 k A Z , X k A Z . Z«A! 



jpsin— _,0 lC os— jp- ,?,tan— — > f 3 cotan -£— ;P 4 vers 



I V VI VII 



K z 2 k h z SB: 2 k K z) ._.. 



— , <p 5 covers — — , f tt sec — — , <p 7 cosec —^—\ (ob) 



which is also an integral of (34), and contains eight independent 

 integers k, /c', .... and nine arbitrary functions %, <p, <p,, .... 



is virtually contained in <p sin — . For though ft, ft 1 , .... 



have no necessary dependence, yet during the time z changes 

 to % + n, which time is simultaneous for each part of (35), they 

 must be relatively constant. Therefore, during this time, cos 



» In a letter to Mr. Babbage, dated Feb. 26, 1824, I committed, througli haste, an 

 oversight, in stating the last term of this solution, which I here take an opportunity of 

 acknowledging. 



