230 Dr. J. W. Nicholson on the Asymptotic 



we shall give the name " asymptotic substitution " to any 

 one of the following pairs of equations : — 



(1) y 1 = R^sinp y 2 = B* cos p 



(2) y Y = S* sinh a y» = S* cosh <x 

 (3)*=»* *=»«-', ... (2) 



where (R S T p a t) are functions of z. 



It appears at once from (1) that if dashes denote differ- 

 entiations with respect to ~, 



y2yi—yiy2=V, (3) 



where C depends on the two solutions chosen. 

 Thus by the first asymptotic summation, 



D1 /R-'sinp , -o* , \ -Da • /R'cosp „x . . \ ^ 



R^cosp ^— 2g|- + R a p'cospJ — R*sinpf — %re~— R^p'sinpJ = C 



or dp C 



In a similar manner it may be proved that 



dz -g' ^~2T ( 5 ) 



Asymptotic expansions for y x and ?/ 2 of any type may, 

 therefore, be obtained when R, S, and T have been found. 

 But writing, in the equation 



y=zy? e", 



dv__C 

 dz ~ u ' 



then on reduction 



uu"-iu !2 +2Iu 2 =:G 2 (6) 



The possioie vames of u are (ip, cr, t), corresponding to 



the values (R, S, T) of v, and making — = (t, 1, J) — 

 respectively. dz 



A solution of (6) is therefore S. Moreover, R and T 

 satisfy similar equations with tC and ^C written for 0. But 

 C disappears on differentiation, and the equation becomes 

 linear, yielding 



u '" + 4:Iu> + 2iiV=0 J (7) 



and (R, S, T) are three independent solutions of this equation, 



