AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 267 
It will now be verified that 
xy {mi# a-[>> aDs^]} 
= ?'/{[>, £] [A a -[a, £][/?, £]} 
= zy{[ a , vm fl}- 
For substituting these values in the first and second ratios of (192), they become 
P[[BM,m] / D [ M> pg ]3 
D[ £, «,£]/ D[ £, « ] 
P[[BM,mi / p[[«],[£]] 
D[ 77 ,«,/ 3 ]/D[ 77 ,«] 
. (193). 
Now, since the equations (16), (17), (18), (28), (29), are equivalent to three 
equations only, it follows that 
DfCBW. [?].W] 
P [ £ * V , « > /3 ] 
= 0, 
which may be written 
r. p1 P[[BW,h]] r .PClBl^im 
L a > PJ n — L a > a J 
p [ I , v , « J P[t,’?>/3] 
, r r n D[[BW>[ri] n 
+ ^^dTTTTTF"] - [ “’ Adc,...^-] - 0 
Also from (16), (17), (18), (28), (29) may be deduced 
which may be written 
P[[g ]»M. [fl W = n 
p [ £ /3 ] ’ 
[/3> .]-D[ ?,,./3] 
+ r/3 -]Pnafei[«] _ r(3 fl punwm] __ 0 
+ LP ’ Vi D [ | , « , /3 ] f J 1) [ v , « , /3 ] “ 
Multiplying (194) by [a, /3], (195) by [a, a], and subtracting, 
P{[flM.ffl} rr ir m ro ir 1 ) 
p [£, « /3] ^-L a ’ ? ?JL a ’Pj [P; i ?][ a J a ]i 
pmid mi 
{[a, £] [a, /3] — [/3, £] [a, a]}, 
(194). 
(195). 
P [v, *, £] 
which proves (193). 
This proves that the first ratio of (192) is equal to the second. By symmetry the 
first ratio is also equal to the third. 
Hence the line of contact is the intersection of the planes 
2 M 2 
