ON TRILINEAR CO-ORDINATES. 65 



From the equations to the line * 



sin 9 sin <p sin jp 



___ ^, ^ (a-a/)2+(^-^/)2-|-(y_y/)2 

 2 sin .4 sin £ sin C 

 since sin 2 J. sin^Q -\- ... =2 sin J. sin ^ sin C. 

 9. To find the angle X between the lines (I, m, n), (I', m', nf). 

 Let (^j, (?2, ^3), (<Pi, ^2} fs) tie the angles made by the lines with the 

 sides of the triangle of reference ; then 



cos 9^ 4- m cos 9,-^-\- n cos e^ = R 

 sin e^-{- m sin 0^ -\-ln sin ^3 = 

 Z' cos 0j + "*' cos ^2 + ^'' cos 0„ = .K'' 

 Z'' sin (j)^ + m'' sin <p^ -j- w'' sin ^3 = 



Multiplying the second and third of these together, and also the first 

 and fourth, and subtracting the resulting equations, we get 



U^ sin dj^ — ^^-{- ....+m?/sin q^ — <j)^-\- m'n sin Q — ^^ + ••• = 

 .-. {IV -\- ...) sinX -{-mnf sin (e^ — 634- ^3 — ^ ) 



+ m^n sin (0^'e^+ e^—'^a) + ••• = 

 which reduces to 



{IV + ... ) sin X + {mn' — m^w) sin — cos \ 



+ {mn' 4- m'n) cos 0^ — 63 sin \ -f .. — 



im'n — inn'\ sin A 4- ... 



.*. tan X = '- — 



IV 4" ... — {m'n 4- mn') cos A — .... * 



November 12, 1870. 



