s 9 S XV. Waring, &c. 
pm is an ordinate in b : for the fines of the angles p'hp, bpp', and 
bp'p ; p'pm, pmp', and pp'm write r, s, and t, r' , s', and t' ; and 
for AP', P 'p', and p'm refpedtively z, x, and y ; then fubftitute 
in the given equation for AP, P p, and pm, their refpedtive values 
z (AP')^— , xjy, x (P'p' — '—,y, and -,y; and there refults an 
y jc it 
equation to the folid expreffing the relation between the two 
abfciffas AP' and P f p' and the tranformed ordinates p'm., 
Fromthefe cafes, which are eafily reducible to one, may be 
transformed any given a-bfciflae and their correfpondent ordi- 
nates into any other containing given angles, &c. with the 
before-mentioned abfciflae and ordinates. 
In the properties of curve lines, firft publifhed in 1762, is 
given a method of deducing the equation to any feCtion of the 
folid, and in particular the cafe of deducing the equation ta 
the projection of any curve on a given plane. 
From the principles given in this, and the Paper on centri- 
petal forces, which the Royal Society did me the honour ta 
print, can be deduced the fluxional equations, whofe fluents 
exprefs the relations between the abfciflfe and their correfpon- 
dent ordinates of the curves defcribed by bodies, of which the 
particles aCt on each other with forces varying according to 
given functions of their diftances. 
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