APPENDIX. 431 



with the ellipse produced by the combination of the forces 



m , ?7i 

 -3- and j-. 



7. This had appeared to be a necessary consequence of 

 the conditions stated, but not as at all proving the ve 

 locity to be the same in the ellipse, when described by 



one force ^ or ^, or when described by the combined 



action of both, or when described by the combined 



r , a m m r , q , 



action of and - L , or of -, o, , and ; be- 

 m m r* q* m m 



cause in all those cases the velocity will be different, and 



^ 



11 i n ?72 ?* . , T)l O .., 



particularly the action of s- H with f + - will occa- 



J ?*- m q* m 



sion adifferent velocity in each point from that occasioned 



by 5- H 5. Thus to take the velocity at one point 



* r 2 q 2 



answering to C. If Ha and Ha be taken as -^ and - 2 , the 

 diagonal lie is the force of ^ and -^ combined, TIC is the 



resultant of and -^-combined (supposing 7?z = l). There 

 fore the velocity in II will be as lie + nC, when all the 

 forces act, and only as He , when the two former act alone, 

 and as nC when the two latter act alone.* But the curves 

 appear to be the same in each case. 



8. These consequences seeming to follow from a con- 



* The difference in velocity is easily obtained, in comparing the effect of 

 one force and of the combined forces, from the equation v- = 2 (fx half chord 



Q TJ 



osculating circle, the chord being = t , p = perpendicular to the tan 

 gent, and R = radius of curvature. 



