238 Mr Brill, On the Orthogonal System [May 21, 
Therefore 
_pdy+ gdé pad — qdry 
(p+ ¢)k, (p+) ky 
Hence, if ds be any elementary arc, we have 
ie da’ a dp” . (pdy + gdd)’ + (pdd — gdy)” . dy* + dé 
hk’ ky (p a Q?) hk,’ (p ata q) k’k, 
dn + du* 
(m? + n’)hoh, 
da and d§ = 
ds° 
Similarly we should have ds’ = 
8. In order to discover the manner in which the theory that 
we have developed for the plane may be imitated in geometry of 
three dimensions, we will seek to determine the geometrical 
meaning of that theory. 
NV 
O M 
n 
Let P be any point, and through it draw a straight line PQ 
of infinitesimal length, in any direction. Through P draw a 
curve belonging to the family whose parameter is 7, and through 
Q draw a curve belonging to the family whose parameter is &. 
Let these two curves meet in R; and let PR=ds, and Qh=ds,. 
Take a length OM on the axis of # equal to PR, and through 
draw MN parallel to the axis of y and equal in length to QR. 
Join ON. Then the elementary triangles PQR and OMW are 
