416 Mr. Bowditch on the motion of a pendulum 
horizontal may be easily reduced to that, where the axis is inclined to 
the horizon by a very small angle. : 
It is worthy of remark, that the distance of the points of suspension — 
AB does not enter into these equations, consequently that distance may 
be varied at pleasure, without affecting the motion, provided rand r' re- 
main unaltered, and the condition at the end of § 2 be observed. 
It may be observed that these equations would remain the same, 
even if the term r’ were to become negative, or the point C were to 
fall above G, provided that by any light mechanical contrivance, 
whose weight might be neglected in the calculation, the ball D could 
be retained constantly in the moveable plane ABC, while vibrating 
about this axis AB, the ball at the same time vibrating in that plane 
about the centre C. 
5. If we suppose the arcs of vibration to be small, the vertical as- 
cent or descent of the body, denoted by the change in the value of z 
would be very small in comparison of a or oe consequently in the last 
of the equations (A) we may neglect a 773 and, as y is very small i = 
comparison of z, we may, in veplectan terms of the order 
2 put [== =I, then this equation divided by dt gives a=g> This 
ttl age af zz Z—+yy eq Vi y $} : 
being substituted in the other two equations (neglecting quantities 
like those before mentioned, by putting r for /rr—axe, and rr for 
= P g ; 
vzz+yy) they become 0=d. +i oon eat O=d. a8 sya which inte- 
r+r 
grated, putting for brevity = —=44,—— =a'a’, give 
ere 
=b cosime (atc) 
=b' cosine (a’t+c’) 
as may be'easily proved by substituting these values in the prope 
equations ; 4, 2',c,¢’, being constant quantities,to be determined by 
(B) 
