GEOMETRY. 



if 'P (r, !,f,) = o be the equation to the fiirface referred to, 

 any three planes, then, in this equation make 



r = x + a, s = y -{- l>, t = a + e ; 

 and an equation of the furface will be obtained in terms 

 of X, V. z referred to three new planes parallel to the firft, 

 and paffing throiigh the point fuppofed to be the centre of 

 the furface : if, by the three particular values affigned to 

 a, b, c, the terms can be made to difappear, in which the 

 fum of the exponents of the three co-ordinates are of a dif- 

 ferent denomination as to even or odd, then the degree or 

 dimenfion of the equation p (r, j, /) = o, the propofed fur- 

 face will have a centre. 



Of diametral Planes. — When, in all the terms of an equa- 

 tion of a furface, the exponent of one of the co-ordinates is 

 an even number, the plane of each of the other co-ordinates 

 divides the furface into two equal and fimilar parts. 



The equation being ^ {x,y, z,) if in all its terms the expo- 

 nent of s is an even number, the plane of x andji will be 

 a diametral, for it will give for % a value a., funftion oi x,y, 

 and conllant quantities, and z = a will fatisfy this equation ; 

 therefore, to the fame values of x zn6. y, two values of a 

 will correfpond, differing only in the lign ; therefore the 

 plane of x,y will be diametral, and for the fame reafon the 

 two other planes of the co-ordinates will be diametral ; when 

 in each term the exponents of x, y are even numbers. 



Let I? (r, s, t,) = o be the equation of the propofed fur- 

 face ; by the transformation of the ordinate, the furface may 

 be referred to three new planes, then Ar-)-Bj-i-C/ + 

 D = o; A'r -I- B'^ 4- C'< -h D' =o; A" r -f B"/ -(- 

 C ''< + D'" = o ; in which equation there are nine conftant 

 quantities. 



The furface propofed has diametral planes, when, by af- 

 figning real and particular values to thefe conftant quantities, 

 the terms in which the exponents of the co-ordinates which 

 are odd numbers, may be made to difappear. The real 

 root of the equation, obtained by making the co-efhcients of 

 thefe terms equal zero, determine the number of diametral 

 planes. 



In confidering furfaces of the fecond degree, great ufc 

 may be made of thefe principles, in determining the centres 

 and diametral planes of thefe furfaces. 



Of Surfaces of the fecond Degree. — Let the general equa- 

 tion of the fecond degree, between three variable quantities 

 x,y, z, he a x' + b y' -{- c -z,^ + dxy + ey x + fxz -\- g x + 

 by\-k% + i — o. To determine if the furface to which 

 this equation belongs has a centre. 



Making ji: = j;' -f a, ^ = _)r' -f- (?, a = z' -f- ">> ^i ^' "/be- 

 ing fuppofed the co-ordinates of the center, the equation 

 becomes a x"- + by'- + c'z'- + d x' y' + e' y' a' + /' a' z' + 

 g'x' -f A ^' + i' z' 4- I = o. 



In this equation, which is likewife of the fecond degree, 

 there are only three terms in which the fum of the exjKiiients 

 of the co-ordinates is an odd number ; thefe terms may 

 be made to difappear, by making their co-efficients etjual to 

 2ero, which gives g' = o ; h' — o; k' = O, making this fub- 

 ftitution and taking only the terms multiplied by x',y'y %', 

 Zaa. + d^+fy + g = 0,2b^ + da + ey-\-h=o, 



Thefe equations being linear in a, /9, y, thefe quantities 

 have real values ; therefore, furfaces of the fecond degree 

 have a centre. 



If a certain relation be eftablilhed between the conftant 

 quantities a, b, c, d, &c. this centre may be placed at an infi- 

 nite dillance from the origin of the co-ordinates. In effeft, 

 the value of »,/?, y are fraftions whofe common denominator 

 k ac' -\- bf-t cd — xab c—def, therefore, when the flol- 

 lowing equation fubfilu between the conftant quantities of 



the general equation of a furface of the fecond degree, tvs- 

 ae' -I- b f^ -f- c d -¥ \ ab c -\- d e f ; the co-ordinates 

 of the centre of this furface are inlinite. The lurface 

 of the fecond degree has likewife diametral planes, for by 

 tranfpofmg the ordinates it may be referred to three new 

 planes, containing nine conftant quantities; taking w, v, -jn for 

 the new co-ordinates, the general equation becomes A xi' 

 ■\- B "a* -)- C tu' -\- Du'U-l-E'UTO-t-Fuw + GK + H'y + 

 K w -(- I =; o ; exterminating thofe terms in which the ex- 

 ponent of any one of the co-ordinates is odd, the iix follow- 

 ing equations are obtained ; D := o, E = o, F = o, G := o, 

 H = o, lv = o; (A). Of nine conftant quantities, fix 

 only are determined by thefe equations ; hence it follows, 

 that three planes may cut a furface of the fecond degree in 

 four equal and fimilar parts in an infinite number of ways ; it 

 has therefore an infinity of diametral and conjugate planes, 

 and of thefe three perpendicular ones, which intcrfeft each 

 other on the three ftraight hues on which are reckoned 

 the axes of the furface. This property is analogous to 

 that of curves of the fecond degree, which have an infinity 

 of conjugate diameters, and in thefe cunes there are two 

 conjugate diameters perpendicular to each other, called 

 axes. The three equations which exprefs that the new 

 planes of the co-ordinates are redlaMgular joined- to 

 the fix equ£(tions (A), determine the nine conftant 

 quantities which enter into the equations of thefe planes. 



Taking for granted what however may be deinonftrated, 

 that thefe conftant quantities have always real values, we 

 may fuppofe, that referring the furface of the fecond degree 

 to its reftangular co-ordinates, its general equation will 

 always be of this form, Lx^ -|- Mjr' + Nz' — 1 = 0. We 

 /hall firft conlider the furfaces comprehended under the general 

 equation, and next the cafe where the centre is removed to 

 an infinite diftance from the origin of the co-ordinates. 



Every furface of the fecond degree interfcfted by a 

 plane, has for its ieftion a curve of the fecond degree ; 

 for whatever be the planes, it may become, by the tranf-' 

 formation of its co-ordinates, one of the planes to 

 which the furface is referred, fo that after this transfor- 

 mation, the equatien to the furface is ftill of the fecond 

 degree ; moreover, the equations of the feftions made 

 on a furface by the planes of the co-ordinates cannot be 

 of a higher dimenfion than the equation of the furface, 

 therefore every furface of the fecond degree cut by a plane 

 has for its feftion a curve of the fecond degree likewife. 



If the interfering plane moves parallel to itfelf, the 

 fcAion remains always fimilar to itfelf : its axes remain 

 always parallel, and its centre is always on the fame diame- 

 ter of the furface, wliich may be thus demonftrated. 



The equation of a curve of the fecond degree may alway* 

 be reduced to this form, 



I X' + my -f n x y -\-p =; o. 



If in this equationy.v and Ai be fubftituted for x and y, 

 y" being a conftant quantity, the new equation which refults 

 from this fubftitution belongs- evidently to a curve fimilar to 

 the fiift, and Cmilarly fituated j it only differs from the firfl 

 in the conftant term, for after having divided all the terms by 

 y-', it becomes 



/*' + my 4- »x_y -t- ---, 



0. 



Therefore all curves of the fecond degree, whofe equations 

 are of this form, differing only in the conftant term, will be 

 fimilar and fimilarly fituated. 



The general equation to a furface of »he fecond degree 

 being « 



• L *'' -}. M »■■' -I- N «' — 1 = o, 



Let 



