342 Mr. Ivory on the Properties of a Line of shortest 
Now we shall find 
av T+e?. of sintl —sin?u 
SS 
Y sin?a— sin? y= 
v1 ezcos?l.1+e%cos?u” 
OY cok ae ee aes 
cosy ~ cosua/l+e*cos?u ” 
ier ne 
cosldu 
we get, -—d¢g= cos u asin’? —sin?u? 
and hence, because cos / = 
gh cos ld 
up a cos27 4 sin2/sin22 ° 
We have therefore, 
do a . cosldsz 1 
cos?/ + sin2/ sin? x A 1 +e (cos? + sin 27 sin® %). 
If this expression of dg be expanded and integrated, we 
shall find, 
sf tang \ ee ai 3et e 
g = arc tan. (=>) 2 x cos 1} = cos Lt. 
; 4 cos 1 si 
In this value I have rejected Be oa fdz sin® z, which 
is altogether insensible even supposing z equal to 10° or 12°. 
Next put 
ee ue cos : L 
2 8 
a 
and the last equation will become 
tan 7 tan z 
We must now find z in terms of z, and, as this operation re- 
quires only the ordinary rules of analysis, I shall suppress the 
detail of the calculation. Neglecting quantities of the order 
already indicated, I have found, 
e 3 r 4 
Re i ; 1+ z (cos*2 + sin?/ sin*a2) — - cos #7}, (D) 
which formula may likewise be written thus, 
z=av 1 +e cos?l) + = sin? Jin? a... - (D’) 
This value of z must now be substituted in the formula for s; 
and, in doing this, it will be sufficient to make 
sm'2'2 = sin2a (1 at <cos*) = sin 2x + e*cos*l x 2. 
Thus we get 
MNT ieas tang cos/ = tan 2, 
