GEOMETRY. 



A (A' C" _ A" C) - B (C B" - C B) = o. 

 By means of thefe three equations the expreflion for L is 



reduced to %'A' + B' + C'. 

 By a fimilar calculation. 



r X f cof. 5 fin. 4- fin. ? + cof. 4- cof, ?) 

 C — 2 fin. 6 fin. (?. 



B '■ + C" ; 



v'A^ 



+ B- + C 

 + B" + C- 



L' = a' A'- + B" + C; L' = VA" + 

 nhich gives for the new co-ordinates u, v, iv, 



■ a = A .T + B^' + C s + D -r 



\. = A' ^ r B' V + C- a + D' -f- 



-.i, = A" .t + B"j- + Cn + D"-^ a'A'" + B'-+_C^- 

 The values of u, i', -j.; mij^iit have been determined directly, 

 fincc thfv are the perpendiculars drawn from the points x,y,%, 

 upon three planes, whofe equations are known. 



It it be required to transform one fyftem of reftangular 

 co-ordinates into another fyftem likewife rectangular, and 

 having the fame origin as the lirll, the three new axes may 

 be given by the equations of the three new rectangular planes. 

 Of the iix conllant quantities which enter thefe equations, 

 three are determined by the condition that the planes are 

 pei-pendicular to each other, and their values are to be calcu- 

 lated from that which is afligned to the tliree others ; but 

 this calculation may be avoided bv determining the polition 

 of the new axes by means of any three angles 4 , b, J. This 

 transformation is ufcful in the application of analyfis to me- 

 chanics. The following method is that ufed by La Place in 

 the " Mechanique celelle.'' 



Let the primitive planes be defignated by two of the 

 three co-ordinates x, y, z, which they contain, and the new 

 planes by tv.-o of the co-ordinates .t'", y'", z". 



Let i be the angle of the two planes x y and x'" y'". 



4- the angle whieh the axis x makes with the line of in- 

 terfeftion of the plane x"' y'" with the plane .r_>'. 



vJ the angle which this line makes with the axis x'". 

 ' It is now required to find the values of x'", y", %'", in terms 

 Oi X, y, «, and of the three angles 4-, ^, ?• 



Let K',y', z', be the co-ordinates of a point referred to the 

 rectangular axes, reckoned upon the three following lines. 

 I. The i'lterfecting line of the plane x'" y"' with the plane 

 X y. 2. The projeiStion of the axis z'" on the plane xy. 

 ?. The asis % ; 



tlien x =■ x' cof I ■\- y' fin, 4- 

 V = y' cof. 4 = k' fin- 4- 



Let x'', y", %", be the co-ordinatej of a point referred to 

 the rectangular axes, reckoned upon the three following 

 lines ; i. The interfering line of the plane x" y'" with that 

 2. The perpendicular to tliis line on the plane x'' 

 The aiis z'" ; thea 



of 



3- 



= y 



the three axes a' ',y 



x"-. 



cof 9 -f- a" fin. « 

 cof. 5 — _y"Cn. 6: 

 being the co-ordinr.tes of tl^e point relative to 

 z'", we have 

 x'" cof. 3 — y"' fin. p 

 y'' =; y" cof. <f> -|- x" iln. P 

 z" = z". 



x" (cof. 8 fin, 4 fin. p + cof. 4 cof. <p) 

 y'" (cof. S fin. -i cof. ^ — cof. 4 fin. ?>) 

 z" fin. 6 fin, 4 



y (cof. 9 cof. 4 fin. 9 — fin. 4 cof. ;) 

 - y'" {cof. 9 cof. 4 cof. f -J- fin. 4 fin. {) 

 + z ' fin. i cof. 4' 

 s = ="' cof. 9 - y fin. 9 cof. : - x" fin. 9 fin. ?. 

 Multiplying thefe values of x, y, z refpeftively by the co- 

 efficients oix'" in thefe values, we have 



Hence x 



-{t 

 = [ 



And by multiplying thtfe values of v,j', z refpeiEtively by 

 the co-efficients of y'" in thefe values, and afterwards by the 

 co-efficients of z'", we have 



r X (cof. 5 fin. 4 cof. 5 — cof. 4 fin. 9) 

 y"< = J -t- j> (cof. 9 cof 4 cof. ? 4- fin 4 fin. f) 



(^ — z lin. 9 cof ;. 

 z'" ;= X fin. 9 fin. 4 -(- J' fin. 9 cof. 4 + z cof. 9. 



Another transformation is fometimes ufed ; a point being' 

 referred to three rcclangular planes by the co-ordinates 

 X, y, z, a ftraight line is drawn from this point to the origin 

 of the co-orduiates ; the length of this line is given, as like- 

 wife the angles which it makes with the three reftangular 

 axes. If )■ reprefent this line, and a,,i, >, the three angles,, 

 then X = 7Cof. ■/., y == r cof. Iz, z =: r cof. -, (i ). 



Of thefe three angles two only are neceiTary ; becaufe 

 cof. a^ -1- cof. $'- + cof. -^ — I. 



When the pofition of a point is thus determined by a lint 

 r and two angles, r is called the radius vef.'cr, and the ori- 

 gin of the co-ordinates becomes a pole, from which proceed* 

 the rat/ii -ve/lores of different points in fpace. 



Sometimes the radius veftor is projected upon one of the- 

 reftangular planes, fuppofe on x y : the angle of the radius, 

 with its projection, is given, as likewife the angle of the pro- 

 jeftion with the axis of .r, tn y, if f reprefent the firlt, and 4' 

 the fecond of thefe angles, 



z = r fin. p ; ji = ; !ln. ; fin. -\. ; x := r fin. ? cof. 4 . ( I ) 



If the point, referred to tlu-ee redtangular planes by the 

 co-ordinates x, y, z, belongs to a furface, we have between 

 thefe three co-ordinates an equation, F (.v,^', z,) = o. If 

 the co-ordinates are transformed, and the new ones become 

 u, V, w, we muft fublUtute in F = o for .r,^', z, their value! 

 in terms of u, v, iu, and the refulting equation will belong 

 to the nev.- furface referred to the new planes. 



If in the equation F = o, for a-.j', z, we fubllitute the 

 values given in equations ( i ) and (2 ), it will become what iS' 

 termed the polar equation to this furface. 



When a curve is given by two equationsy" (.i-,^', z) = o,- 

 f (.r, ^, z) := o, in fubftituting in thefe equations the values 

 given by equation F (x,y, z) — o, we obtain an equation to 

 the curve, relating citlier to three new planes by the co-ordi- 

 nates II, V, 'uj, or to a pole, by the rmlU veSores ,^TiA their angles.. 



Of the centres of furfaus, and of thar diametral planes. — 

 The centre of a furface is defined to be a point, in which- 

 all the chords paffing tlirougli this point are divided into two 

 equal parts. 



A diametral plane is that which divides a fyftem of parallel 

 chords, each into equal parts. 



Hence, if a furface has a centre, all the diametral planes 

 which it can have, neceflarily pafs through this centre. 



Having given the algebraic equation of a furface, to deter- 

 mine, 1 ft, if it has a centre ; 2d. if it lias a diametral plane. 



if the propofed iiirface has a centre, let it be referred to 

 three planes, the origin of wliofe co-ordinates is the centre 

 itlelf. 



Any ftraight line drawn through the origin of thet? co- 

 ordinates will be a diimicter, and will cut the furface in two- 

 points, the co-ordinates oi the firft being x,y, z,and of the 

 fecond — .v, — y, ~ z. Therefore, the equation to the fur- 

 face mull fubfill iu taking .v, ji. z, pufitive or negative ; to 

 fatisfy this condition, the fum of the exponents of the three 

 co-ordinates in every form muft be the fame in every 

 parity as the number which expreflcs the degree of the 

 etyiatioD piopofrd, that is ; if tvco, even, if odd^cdd. Thuf, 



if 



