VOL. LI.] PHILOSOPHICAL TRANSACTIONS. 445 



-7=7 IS = ^'P + TTT + 



VZ\ '" — '^^ ^ 2.3 "^ 2.2.3.4.5' 



— ^= = Z)P 4- 4- ^- 



^/Z7 — i^^ T 2.3 "f 2.3.4..5 "I" 2.2.3.4.5.6".7» 

 &C. &C. 



Whence the values of p", p", &c. ql\ ql'\ &c. a", a'", &c. may be easily ob- 

 tained, in terms of a. 



Q. Hyp. log. CyztJ^ being = a; + |' + |, &c. 



g' = fluent of ^ hyp. log. Cjzr}^ is = ^ + | + |!, &c. 



g" = fluent of ^ g' = 07 4- |J 4- !„ &c. 



g'" = fluent of ^ g'' = a; + J-' + ^', &c. 



X '3^ 5^ 



&C. &C. &C. 



10. By writing, in the first equation in the preceding article, - instead of x. 



we have 



1 

 1 +- 



ar-* , x-' 



Hyp. log. ( L)4 =:.-» + 1^ + :::^, &c. 



1 — 



X 



1 



1 +T 



But the hyp. log. of ( — i)3 is = hyp. log. (^~)^ = hyp. log. (}^-^)^ q: 



1 



X 



hyp. log. s/ -I— ±b-^ hyp. log. (J-^-^) . 



It is manifest, therefore, that 



Hyp. log. (14^)* is = + 4 + ^' + ^ + fl', &c. 

 where, with respect to the two signs prefixed to h, the same observation may be 

 made as in art. 3. ' 



1 1 . Multiplying the last equation by - and taking the correct fluents, we have 



g' = iQi" + Z)X — a?-' — —, &C. 



Whence, by multiplying by -, and taking the fluents, we get 



G" = 2a"x + % + ^-' + ^ + ^, &c. 



Again, multiplying the last equation by -, and taking the correct fluents, we 



find o'" = la- + Q^x^ 4- ^^ - ^~' - ^' - 9^ ^^• 

 And, by proceeding in the same manner, we find 



&c. &c. 



