242 THE REV. ROBERT HARLEY ON A CERTAIN CLASS 



to the ordinary form of a differential equation, we find 



2^ . 3M5*-^)^ + 2^ . 3^(2^ • 5^-23«^)^ 

 + 2.3^(25.3^5^-1249^)^ 



+ 3^(2^3^5'-ii.i8i^)g 



+ 2(2^ . 2^ — io^ioc)-^ + 2y = 2. 



The accuracy of this equation may be tested as follows : — 

 From the trinomial sex tic we deduce, by successive dif- 

 ferentiations. 



dx 6 * y\y-s) 

 ^> 5^ 2/ -4 



dx"-- 6^'y^y-sY 

 ^__5^ iiy^ — 88y+i8o 



d^y _ 5^ 1872/' — 2244-2/* + 9 140^—1 260a 



dx^- ~6"4 • y'\y-sy ' 



dx^ 6^ 



430I2/''-— 688 162/^ H- 4192402/* — 1 1502002/ + 1 197000 

 • y^\y-sf • 



If now we assume y=i, then we have a7= i, and the 

 above values give 



dy^ 5 ^y^_JL_, 



dx 2^ . 3' dx"" 2^ . 3^ 

 6?V_ 5^.io 3 d^y^ 5^-613 



dx^ 2^^3'^ fl?^^ 2'^3^' 



d^y _S^ . 1 606 1 

 ^^""2^^755" 



.> 



which, substituted in the foregoing resolvent, are found to 

 satisfy it. 



