Tlie Rev. Brice Bronwin on the Theory of Planetary Disturbance. 59 



The term in N containing z is of the order of the third power of the dis- 

 turbing force multiplied by sin' /, and cannot, I conceive, in any case be 

 wanted. 



"We may transform the co-ordinates x', y', and z', of the disturbing body 

 ni like manner to a sliding plane, and to a similar situation on that plane. 

 Thus, marking all the corresponding quantities relative to this body with an 

 accent, we have 



x' = x\ cos {e' - a'n't) - y\ sin {0' - a'n't), 



y' - w', sin (0' - a'n't) + y; cos (0' ~ a'n't), z' = z',. 



.?■; = x'^ , y', = y'^ cos i' - z'., sin f , z[ = y'^ sin i' - z'^ cos i'. 



x', = x', cos (tt' + Q'n't) - y'^ sin (tt' + C'n't) , 



y'2 = x', sin (tt' + t'n't) + y', cos (tt' + ^'n't), z', = z',, 



But we may transform x', y', and .-' to the plane of the orbit of the disturbing 

 body, if we please. 



To find the latitude, marking the letters in the second member with the 

 number of the transformation to which they belong, we have 



z = y, sin i + z^ cos i = x, sin i sin (tt -1- ^nt) -f y^ sin i cos (tt -l- ^nt) ■\- z^ cos i. 



But <7 being the sine of the true latitude, s the sine of the latitude relative 

 to the sUding plane, and/ the anomaly, or the longitude measured from the axis 

 of X after the last ti'ansformation ; 



z = r<s, xs-rcosf, y3 = rsmf, and Zs = rs. 

 Therefore, by substitution in the last, and dividing by r, 



a = sin z j sin (tt -I- Snt) cos/-l- cos (tt -H Snt) sin/| + s cos i ; 

 or, 



<^ = sini sm(f+TT + Qnt) + scosi. (6) 



Thus the latitude is very easily found. 



Still marking the letters according to the transformation to which they 

 belong, 



yi = y. cos i - z. sin ; = ^3 cos i sin ( tt -f Qnt) -\- y^ cos / cos (tt -(- Qnt) - Zs sin i 



i2 



