Royal Astronomical Society. 387 



of which we have spoken, namely, a force which acts in opposite 

 ways on opposite sides of the moon's orbit, and in which the period 

 in the change of the nature of the force coincides nearly, but not 

 exactly, with the time in which the moon moves from apogee to pe- 

 rigee ; for then the pulHng force at apogee will after a long time be 

 changed to a pushing force, and the pushing force at perigee will in 

 the same time be changed to a pulling force. If, for instance, the 

 change in the disturbing forces of Venus (from pushing to pulling) 

 occupied fourteen days exactly, and if the moon's motion from apo- 

 gee to perigee occupied fourteen days and five minutes, then in 4032 

 anomalistic semi-revolutions of the moon (which would bring her 

 from apogee to apogee), there would have been 4033 changes of the 

 force (which would change it from pulling to pushing), and there- 

 fore in this time, and no sooner, a complete pulling force at apogee 

 would be changed to a complete pushing force at apogee. 



It is necessary now to point out how such a modification of the 

 force of Venus can be found. 



The only disturbing forces which are yet completely brought under 

 the management of mathematicians are of two kinds ; a constant 

 force (always pushing or always pulling with the same amount of 

 force), and a force alternately pushing and pulling, having equal 

 periods and equal maximum magnitudes in each state. The latter 

 of tliese, if projected graphically, with the time for abscissa, is re- 

 presented by the ordinates of a line of sines : algebraically, it is ex- 

 pressed by tt.cos (^bt + c). 



Now, while the relative positions of the earth and Venus change, 

 the disturbing force on the moon (estimated by the force which, on 

 the whole, it exerts to pull the moon from the earth) undergoes very 

 great changes. When Venus is nearest to the earth, this force is 

 about 250 times as great as when Venus is furthest from the earth. 

 It declines very rapidly from its greatest magnitude. If therefore 

 we represent the disturbing force from one conjunction to the next 

 by a curve, this curve will be very high at the beginning and end, 

 and very near the line of abscissa at the middle, and through the 

 greater part of its extent. 



The separation of this foi-ce into a number of different forces, fol- 

 lowing the two laws mentioned above, is effected by a process sug- 

 gested and facilitated by algebra, but in which, nevertheless, every 

 step has its physical meaning. It may be stated at once, that this 

 remark applies universally to the algebraical operations of physical 

 mathematics. As a simple instance, we may refer to the equation 

 (a + bysz a3 -J- 3 a"^ b +3 ab'^ + b^, which probably was suggested by 

 algebra ; but which may be illustrated by taking a cube, whose side 

 is a + b, and (by three saw-cuts) cutting it into eight pieces, when 

 the single piece representing a^, the three pieces each representing 

 a- b, the three pieces each representing ab^, and the single piece 

 representing b^, will be found. And there is perhaps no better dis- 

 cipline for the mind than thus tracing the evidence of the truth of 

 algebra, especially in its more profound processes. 



The separation, then, of the force of Venus goes on by the fol- 

 lowing steps : — 



2C2 



