362 l-HILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



circular arches and hyperbolic areas, or between the measures of angles and of 

 ratios, as the only principle on which the imaginary investigation can proceed. 

 It need scarcely be observed, that the exponential value of the hyperbolic ordi- 

 nate may be deduced from what has been proved in this article. 



J 1. But as the arithmetic of impossible quantities is no where of greater use 

 than in the investigation of fluents, it is of consequence to inquire, whether the 

 preceding theory extends also to that application of it. Let it then be required 

 to find the fluent of the equation 4^ + d^y = q, where q denotes any function 

 whatever of ar. For this purpose, the following lemma is premised: let a' be any 

 arch, and/) any flowing quantity; then, if the sign /, be taken to denote the 



fluent of the quantity to which it is prefixed, sin. x J p cos. .r — cos. x /p sin. x 



=: ., /_i /p^~'^^~ ' ^ZT I V^^'^~^'-' *^'' 'f \^ be a hyperbolic sector, ord. 



X I y abs. X — abs. X /p ord. x = —/ pc—^ ~ /p^- 



Because sm. x / p cos. x = — — / p X — - — — , by sepa- 



rating the terms we have sin. x / p cos. .r = ^''^_ / pc^V—^ -\- ^1^— /p 



c—xy—i T~7irr I V^^'^~^ — I pC—xV—^, for the same reason — cos. 



xjp^iu.x = - -—-Jpc^V-^ + -—Jpc-^V-^-—^~Jpc^^-^ + 

 ^Zl2:— I c-'^^—^. Wherefore, by collecting the sum of all the terms, we 



have sin. x Jpcos. x — cos. xj p sin..r = 'tttzI J pC-xV-^ — ■ „ '/_ , ' /pC^^-'. 



The demonstration in the case of the hyperbola is free from imaginary ex- 

 pressions; but, in other respects, is exactly similar to that which has now been 

 given in the case of the circle. 



12. Let the co-efficient of y in the proposed equation be first supposed nega- 

 tive, that is, let ^ — a'y = Q, and if we multiply by c^^.v, it being a constant 



but indeterminate quantity, it becomes — :=- — a^c'^y.v = c'"''a.i: Let c"' X 



(^ — By) be assumed for the fluent, a and b being indeterminate, and let its 

 fluxion be taken, then, '' 



— — -f- 7iAc"''y — nBc"'y.i' = c'-'ai". 



i - — BC"*^. 



Hence, by comparing the terms, we get a =: 1, ?2A — b = O, »b = a'; there- 

 fore, n = + a, and B = + o: for n and b let the value -f a be substituted, 

 and for a, its value, unity; then the assumed equation becomes 



{-. — ay) X C* = /c" Q,r, or^ — "j/ = ^~'"' / c-^a.i. Let this equation be 



