56 Mr. Mac Cullagh on the Laws of 



cos(i.+.',)=cos(«.+ 0-(''-0 (41) 



nearly, since «,+ '2 ^i^^ J^ot differ much from a right angle ; and because 



sin tj=5 sin tj, sini'^^ssint^, (42) 



we shall also have, rigorously, 



sin^'^-sin%=(/-^')sin^ =(a'-i'Osin'w'sin^,, (43) 



or 



. ,, V /mtn ,„v sill W SlU t, /1A\ 



which may be written 



,',-,,=(a'-5') iillj^, (45) 



with sufficient accuracy. This value of t'^— t^ having been substituted in (41), 

 the resulting expression for cos (i^-\- t'J must be substituted in equation (40), 

 which will then become 



cos((,-i-O(tan0 + cotane) — (a^— J^)sm\sma)( 7-5 h-^-z — ^1=0, (46) 



if, denoting the arc Ko by w, we confound w' with w, & with 0, and write cos2<2 

 instead of sin(<j — j'„). Multiplying all the terms of (46) by sin0cos0, we find 



cos('.+ 0=(«-^)sin\sinc.cose(^ ^^^^^^ +-^^)- C47; 



6i 



and put R Y=jp, AR= 5 



COS(U = COS5'COS(j9 — ij,) 



From A draw the arc AR meeting the arc Pi at right angles in the point R, 

 / I and put R Y=jp, AR= q. Then by means of the values 



sin w cos = cosy sin (p — t^), j ^ 



afforded by the right angled triangle ARo, the equation (47) will take the 

 ^ ji/} ^°^'" 



^ or 



"^K fL f ^»"f//f) cos{..+ .,)=Kcos'}tsin>-sin'.,), (60) X. 



