134 DEMONSTRATIONS of &c. 



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

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

 Ez, be drawn to xy, xz, 4AD 2 .EP 2 +4AD 2 .EQ^-f-4BD 2 .ET 2 + 

 4BD 2 .EN 2 +4CD 2 .EZ 2 +4CD 2 .E^ 2 = 

 8(AD 2 -fBD 2 -f-CD 2 )(Ey a -fEz 2 )+0 4 . Therefore, 



S 4 - 3DE 4 +A i (Ey 2 +E2 2 )+^ 4 +K AD4 + BD4 + c D 4 ). Or, 



S 4 - 3 DE 4 +A 2 (Ejk 2 +Ez 2 )+B 4 . 



Therefore xy, 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, l$c. and let 

 a, b, c, &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 drawn perpendiculars Ey, Ez, to the two lines found, and 

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



AE2+ A BE2+ ^cE 2 &c.= fj ^^DE 2 +A 2 (Ey t 4-E2 ; 2 HB 4 . 



The invefligation is perfectly fimilar 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 Orations is no way 

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

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

 lue be affigned to it. 



XIII. 



