[929 
929] 
APPLICABLE UPON A GIVEN SKEW SURFACE. 
235 
z) to the 
d by the 
have 
,nd L x to 
called S, 
We have thus the five equations 
a 2 + /3 2 + 7 2 =l, 
x 2 + y' 2 + z' 2 = 1, 
a'x' + fi'y' + 7 V = 0, 
ax + /3y' + 7z' = cos to, 
sin 2 to 
a' 2 + /3 2 + 7 2 = —r—; 
T“ 
and if we herein consider to and r as denoting given functions of s, all the skew 
surfaces which satisfy these equations will be surfaces applicable one on the other. 
Adding to the foregoing the derived equations 
aa + /3/3' + 77' = 0, 
x'x" + y'y" + z'z" = 0, 
ax" + fty" + 7z" = — sin to. to', 
ax'" + /3y'" + 7z'" + a!x" + ¡3'y" + 7’z" = — sin to . to" — cos to . to' 2 , 
we find without difficulty 
0' _ $ z> ~iy yx'-az' a y'-ftaf 
’ 7 T ’ T T 
O ' 0' ' ' a> , n — x' + a cos to —y' + 8 cos to — z' + 7 cos to 
/37 - /3 7, ya - 7 Of, a/3 - a /3 = z , - ^ 5 , f . 
Putting, for shortness, 
and, as above, 
we find