! 20 ■ 
Mr. Land^n’s Invefugation of 
fluents, we have 
a~ x 
D itf — C' 
Dr 2 
and f 
a - X 
D, 
.By* 
'£ 
Dr 2 
£ 
m and n denoting the values of s and t , when g Is — a and 
7 = 0 ; and B being put to denote the difference D - C = d z - c z . 
If now (3 and a be put to denote the colines of BP and DR 5 
tp the radius.#, we dial], from what is done above, have 
JS = - = 
a 
^ . 
0 - 
D 2 
1 
D 2 
ft + y 2 -r o 2 =? (3 {3 -j- 77 + f'i — o ; 
b*ft + c z y +,d z 3 z ~b z n z -p d z m z , and b z {3[3 + c z yy + d 2 l^zz o. 
Drawing AR fo that. D 2 xdine of BR fhall be = C 2 <?, it is; 
very remarkable that the momentary pole (P) will run round;, 
about the point-B 7 or about: the point D, in the fpherieal furface,. 
according as the initial pole fhall be in the part BCR or, DCR 
of the faid fur face ; that is, according as Y)m z is lefs or greater 
than Ca 2 1 and that, if the; initial pole (P) be any where iii. s 
the great circle CR,. the momentary, pole, keeping in the arc 
of that circle j. will: continually approach: nearer and: nearer to. 
the point C in the furface of the fphere ; but, by what fol- 
lows, we fhall And that it never can arrive at that point; in any 
finite time !* 
The equation of the track of the : pole in^the projedtion to, 
which we have hitherto referred will, it. is now obvious, bee 
v 2 — x x 2 + . ■ T h- ; at, me.afured from the. center, A upon 
AD,, being —l ;.. and y,. at right angles thereto, = /3.-. 
If C be = 0 (that is, if c he = 3), v x will be equal to the in- 
variable quantity m ; the projedted track, a right line parallel to , 
AB; and the track upon the furface of . the fphere, a lejfer 
cjrcle in a plane parallel to the plane of the great circle BC„ 
5 ' : k 
