52 Scientific Proceedings, Royal Dublin Society. 



which may be written 



[(/3 - nY - X 2 ] «* + [(« - IT' - -K 2 ] f + [(« - S) 2 + O - ») 2 - &} *' 



-2(a-^)(/3-r,).r?/ + 2[-^{(/3-^) 2 - J B 2 } + ^a-^J{j3-„) + (a-e)i? 2 ].r 

 + 2[^(a-^)(/3- }? )-r,((a-^-^ 2 }+(/3-r,) J R 2 ]y+[{(/3- J ,r--^ 2 j^ 



+ {(a-^) 2 -i2 2 }r, 2 -2(a-S)0-r ? )^-2(a-^^^-2(/3-»,)^'> 



-i2 2 {(a-S) 2 + £S-,,) 2 }] = 0. (1) 



If m denote the distance of the moon's centre, and r its radius, 

 the equation of its surface is 



(x-u) 2 +y 2 + z 2 = r 2 . (2) 



Writing the equations (1) and (2) for convenience thus — 



ax 2 + by 2 + cz 2 + 2 hxy + 2lx + 2my +d=0, (3) 



and x 2 +y 2 + z 2 + 2l'x + d'=0, (4) 



and eliminating # between the equations (3) and (4), we get as the 

 equation of the projected curve 



[a (y 2 + z 2 + d') - (by 2 + cz 2 + 2my + d)J + 4 [hy + 1 - al'~\ x 

 [(% + I) (y 2 + z 2 + d') - l'(by 2 + cz 2 + 2my + /)] = 0, 

 which may he written 



[(« - b) 2 + 4 A 2 ] if + (a - c) 2 z i + 2[(a- b) (a-c) + 2/r] y 2 z 2 

 + 4[h(l- al' + l-bl')-m (a - 6)> 3 + 4 [_h (I- ar+l-cl')-m{a-c)~]iiz 2 

 + 2 [(« - b) (ad' -d) + 2h (hd'~ 2mV) +2(1- al') (I- bl') + 2m 2 ] y 2 

 + 2[(a-c)(ad'-d)+2{l-al')(l-cl')]z 2 

 + 4 \_h (Id' - I'd) + (l- al') (luV - 2ml') - m (ad' - d)~] y 

 + (ad'-d) 2 +4(l-al')(ld'-l'd) =0 . . . . A. 



Let p = the sun's parallax, 



s = the sun's angular semidiameter, 

 zj = the moon's parallax, 

 p = the moon's angular semidiameter, 

 a = the angle of the cone, 

 and B = the angle between the axis of x and the axis of the 



cone. 



