VOL. LXXXVIII.] 



PHILOSOPHICAL TRANSACTIONS. 



417 



- V(i - yy) _ 



we shall have 



3^/(1- yy) 

 4yJ 



+ 



3m 

 4>y 



C " 4.6.8.' 



~ 4.6.8.10.6 "*" 



3. 5.7. 9- H j/ 7 

 4.6.8.10.12. 8* 



&C. 



the sake of brevity, there will be y 



2y b ' 2y 3 \6y l6 



And, if this equation be put into fluxions, and a be written for v/(l — yy), for 



5 ■ 9 j_ _3_ _ 



J i 3m __ 3wy 



Q 



2J/4 4^y " 



6 T" 4y 4 T 40 f 

 _4_ j6_f 



9« 4 4vv / 



4j/y 



$& ■ 2y< 



j3_ 



4.6.8.4 



+ 



4.6.8.10.6 



^ + 



4 6.8.10.12.8 



&C. 



And this equation, more concisely expressed and divided by y, gives 



v 2y ' 4_y 5 



+ (— -- 



3uy 



4y i ' Q 

 5 5 v • 



i6j/ s i6y y ~ 



3.5.7. 3yy , 3.5.7. 9-5yy* 



T 



3.5.7.9.1 i .7jy» - 



4.6.8.4 ' 4.6.8.10.6 ' 4.6.8.10.12.8 



Now the fluent of the series on the 2d side of this equation is found, by the 

 methods which have been long known, to be 



3.5.7.3j/y , 3.5.7.9-5y 4 



, 3. 5. 7. 9. 11. 7y 6 



\ A (^ o in in o £> 



&c. and the fluent of the terms on 



4.6.8.4.2 ' 4.6.8.10.6.4 ' 4.6.8.10.12.8.6' 



the first side will be very easily obtained, by the following assumption, and attention 



to what was shown in art. 2 of this paper. 



For the fluent of the terms on the first side of this equation, assume 



+ 



(£ + £ + £)* + "(£ + *) 



P 



-|f- i% t-j [- s h. l. y ; then will the fluxion of this expression be 



— 6a 46 2c 



f 



y y 



5a 3b c_ 



T f y l y 

 T y % y. 



, ( z^p _ i? __ §r _i_ i. 



f 



4q 2r _£_■> 



-4-4 



y, which being put = the first side of the foregoing 



equation, there will arise as many simple equations for determining the co-efficients 

 a, b, c, &c. as there are letters of that kind in the assumed fluent, from which 

 their values will easily be found. The variable part therefore, of the fluent of the 

 first side of the above equation, is 



,— 5 7 5 x . / 3 . 



a (- 



\6yy' \ K 8yy 



I jj \ t j _j ^_ 



12v° ' 8y4 32jcy" 



16^' ' " y 8yy ! 16' ' |%* n 8y* " 3f#" N ° W ' t0 disc °ver 

 the constant quantities which lie concealed in this expression, we must proceed as 

 above in art. 3, whence is obtained, 



VOL. XVIII, 3 H 



