508 Journal of the Asiatic Society of Bengal. [November, 1908. 
asymptotic angles are suppliementary, then we can prove in the 
same wa 
Q,P. Q,P = QP? (91) 
Again, if (a, >), and (a2, b,) be semi-axis corresponding to 
A, and Ag, then by (72) 

at 27Q4 
a, b= =i a,b,= si 
A? 
972 Q'P® 
Therefore, a) by as b= (9Qe+ (3QQ, - BP)*}8 
=a* (92) 
where a is the semi-axis of the osculating equilateral hyperbola. 
24, par system of simple binomial differential ae 
P,Q, R, 8, T, Q, RB, 8’, ae perk been introduc 
preceding iar acon o urse, be taken with any 
independent variable. Of fier sights quankiises only the first five 
may be looked upon as primary, and the rest as dependent 
constant, and gear dx, dx, d*x, déx all vanish. 
quantities P, Q, R, 8, 7’, Q, are, in this case, equal to (1 ‘pda! 
qdz®, rdz', sd’, taal, pqdae’, respectively. R’ and S’ evidently 
vanish, 
If we take the arc (s) as the independent variable, then 
P=dz? + dy? =ds? = constant 
Therefore, Q, =dad’x+ dyd*y =4dP =0 
9 B+ Q? @ 
it Giet 2  ~. 93 
Again dQ, =(d’x)?+ (d*y)?+dx de+dy d@y=; @P=0 
2 
Therefare, dx d®x+ dy déy = -< (94) 
Also, dz R’ - dz R+ dz Q=0 
dy B’—d*y R+d*y Q=0 
Therefore PR’ — BQ, + (dz d’z+ dy d’y) Q=0 
Hence R’ -< (95) 
Aas: s'=an=*2* (96) 

The general differential equation (54) of the conic, if s be the 
independent variable, therefore, becom 
10 1+ 9QT=45 QR (:-£) (97) 
