184 Pfof. Gayley, On the transformation of coordinates. [Oct. 28, 



Drawing a figure, it is at once seen that 



v^ = ad + 71 -a' x/i^^' • cos [B -6'); 



where cos 6 = , — . , and therefore sin ^ = 



cos a =■ . ,— =^ „ sin^ — 



the values of v> v'. 



^ =l-a^ -13' -v' + 2al3v, 



the resulting value of v^ is therefore 



v^ = aoi + -J-— 5 {(/3 - av'^^' - av) + ^/vv'i• 



The equations 



«: = (a, b, c, f, g, h][a, ^, 7)', « = (a, ...\d, /3', 7')' 

 give (got + f/3 + C7) ' = /f V, 



(ga' + f/3'+C7y=A^V': 

 and we therefore have 



{goL + f/3 + C7][ga' + f/3' + C7 ) = iiVv V' ; 

 recollecting that 1 — i;^ = c, the formula thus is 



v, = aa' + -\{j3- av\iB' - av') + -^ (ga + f/3 + c^^ga + f/3' + 07 )| , 

 or say, 



Kv^ = Kai + - {/v (/3 - avjp!- dv) + (ga + if3\gd + f/3')} 

 c 



+ g (^^7' + <^'7) + f (/^7' + /3'7) + 077'- 



The sum of the first and second terms is readily found to be 



= aota' + b/3y3' + h (a'/3' + a'/3), 



and the equation thus- becomes 



Kv^ = (a, b, c, f, g, hja, ^, 7][a', /3', 7'), 



as it should do. 



