Spherical Motion . . 535 
be the equation of the curve when orthographically projected 
upon the plane of the great circle CA. For let the fphere be 
thus projected, then the quadrants AB, BC (fig. 9.) will be 
projected into the right lines BA, BC, and if Z be the pro- 
jeCted place of the fixed point at any inftant, let fall the 
right line ZX perpendicular to BC; then, by the nature 
of the projection ZX = cof. AZ, andBX = cof. CZ, and if 
^ b z t-\ 1 c z 
C, the above equation becomes 
b'-i 
=A, 
2 2 
a —c 
B, and 
2 L 2 
— 0 
■*A 
BX* = — x ZX'" 
O c 
m 
’SAW 
4 - /»VC% and ZX' 
X 
\X Z + 
m 
C ' ' 
«VC) the projected track therefore upon the plane is 
an hyperbola, whole centre is B, abfcida BX, and ordinate 
ZX, and taking ZX = o, BX = me' C <T - — = the diftance 
c 
from B at which the curve cuts BC, and is therefore the femi- 
tranfverle axis of the hyperbola. But this is only poffible 
whilft CC 2 is greater than AT ; for if CC 2 = AT, XZ = BX x j 
T J — , the projected track is a right line BU, and the real one 
a great circle of the fphere palling through B. If AT be 
greater than CC 2 the track will no longer cut CB, but mull 
cut BA, and BU will in both cafes be an afymptote to the 
projected track. Since the track in all cafes erodes the great 
circle CA, and we are at liberty to fuppofe the motion to begin 
at what point thereof we pleafe, it may be fuppofed to com- 
mence where the track erodes CA, and where, of confequence, 
the velocity along CA is then — o ; we may therefore take the 
adiimed quantity 05 — o, and dill all the conditions of the pro- 
blem be fulfilled, the expredions thus becoming more Ample, 
4 . for 
