134 DEMONSTRJT'rONS of, 8cc. 



point E, the perpendiculars EP, EQ^ET, EV, EZ, Ea, be drawn 



to KP, KQ^LT, LV, MZ, M^, and if the perpendiculars Ey, 



Ez, be drawn to xj, xz, 4AD^EP'+4AD^EQ^+4BD^ET^ + 



4BD^EN'+4CD^EZ = ^-4CD^Efl^ = 



8(AD'+BD^+CD')(Ej^+Ez = )+a*. Therefore, 



S* = 3DE''+A'(Ej'+Ez=)+a*+i(AD*-fBD*+CD''). Or, 



S-* = 3DE*+A^(E_j''+E2;*)+B*. 



Therefore xj, xz, are the lines, and D the point, required to be 



found. 



THEOREM XXVIII. 



Let there be any number, m, of given points A, B, C, ISc. and let 

 a, b, c, i^c. be given magnitudes, as many in number as there are given 

 points, two Jlraight lines, zy, xz, may be found, which will be given 

 by pofition, and likewife a point D, fuch, that if from any point E, 

 there be draivn perpendiculars Ej, Es, to the two lines found, and 

 if EA, EB, EC, ED, be joined, 



AE'+-rBE^+vCE'&c.= ^^^^=^^^*DE«+A'(E;''+Ez')+B*. 



The inveftigation is perfedlly fimllar to the former ; only the 

 point D is not the centre of gravity of the points A, B, C, &c. 

 but, as in Theor. 7. the centre of gravity of weights, fuppofed 

 to be placed in thofe points, and proportional to the magnitudes 

 a, b, c, &c. 



The univerfality of the preceding demon ftrations is no way 

 affeded by our having always fuppofed m equal to fome parti- 

 cular number, becaufe the reafoning is the fame, whatever va- 

 Ivie be affigned to it. 



XIII. 



