172 Miss Dorothy Wrinch on an Asymptotic 



is therefore, P_, Q, R, S being constants, 



P A 4 (l + <*, 1 + 6, 1 + c, 1; *) 



+ ~-«Q A 4 (-a, 1 + 6 — a, 1 + c-a, 1; s) 



+ *-*R A 4 (-&, l + c-6, l + a-6, 1; s) 



+ 5r- c S A 4 ( — c, 1 + a— c, 1 + 6-c, 1 ; *). 



Let y 1 y 2 y$ y 4 represent these four solutions, which together 

 form the general solution. Then the results obtained with 

 respect to the asymptotic behaviour of 



A 4 (l + «, 1+& 1 + 7,1 + 8; z) 

 give 



+ e~ 4L ^ + ^ (a+6+c + 8 / 2) +e- 4 ^+^( a + 6+c+3 / 2 )l, 



Tl-aTl + b-aTic-a _«±A+£±Mr , i , 4** -~(-3«+6+,+3/2) 



^ 9 5/2 3/2 -* 4 «** +« ^ " 



+ ...], 



2^ 9 5/2 3/3 4 |« +« <? 2 



^ 7T 



ri-ari + a-cri + 6-c _ M 6+^+3/2 -. 



3/4 " 28/2^/2 -3 4 [^ j . 



Linear combinations of y Y y 2 y z y^ therefore exist which 

 behave asymptotically near infinity like 



(g+6+c+3/2) i «4-/ ; _)_ c+ 3/2 x 



_ g+6+c+3/2 «4-i +c+ 3/2 



^ 4 cosl^i, or c _ 4 "sin 4s*. 



Let us consider the linear combination which is asymptoti- 

 cally equivalent to 



a+6+c+3/2 a i 

 2 4 a" 4 **. 



