522 FOURTH REPORT — 1834. 



Professor Hamilton finds for this inverse problem the formulae 



^ - ^^TF' ^ ~ a^ + ^^ ' ^ ' 



in which a |3 are the constants deduced as before by (r.) from 

 the base-pair («, b), and involving the integer i in the expres- 

 sion of /3 ; while p and 9 are deduced from u and v, with a new 

 arbitrary integer k in fi, by expressions analogous to (r.), namely. 



=/ 



'/u« + «'V/r 



(v.) 



fl = 9o + 2 ^ TT, J 



in which Aq is supposed > — -rr, but not > tt, and 



cos dn = — T-: ;; ' sm d^ = 



^„2 + j;2 " ^,,2 ^. 



(w.) 



By the definition of quotient, which the definition (d.) of pro- 

 duct suggests, the formulae (u.) may be briefly comprised in the 

 following expression of a logarithmic pair : 



(-■^) = fei^ • • • • w 



and, reciprocally, the direct exponential pair {u, v), as already 

 determined, may be concisely expressed by this other form of 

 the same equation, 



ip,&) = {x,7/)icc,^), (y.) 



if we still suppose 



{u,v) = (/p . cos9,/p . sin 9), 



{a, b) = {fee . cos /3,/a . sm /3). . 



Thus all the foregoing results respecting exponential and loga- 

 rithmic pairs may be comprised in the equations (y.) and (z.) 



When translated into the language of imaginaries, they 

 agree with the results respecting imaginary exponential func- 

 tions, direct and inverse, which were published by Mr. Graves 

 in the Philosophical Transactions for 1829, and it was in me- 

 ditating on those results of Mr. Graves that Mr. Hamilton was 

 led, several years ago, to this theory of conjugate functions*, as 



* An Essay on this theory of Conjugate Functions was presented some years 

 ago by Professor Hamilton to the Royal Irish Academy, and will be published 

 in one of the next forthcoming volumes of its Transactions. 



