BOSCOVICH'S THEORY. 



155 



Boscovich's XII. If now we make TQy=0, this equation 

 Theory of W JU f u [gj a u tae conditions proposed but the last ; 



Pha Ural anCj l ^ at mi Y oe * l, "^' et ^ an '"finite number of 

 p] lv . * ways, by properly determining the value of Q. 



- ' For, in the first place, since the values of P and 



Q, put equal to 0, have no common root, they have 

 no common divisor, and therefore the equation can- 

 not by division be reduced to two ; it is therefore 

 simple, and expresses one simple and continued curve, 

 which is not composed of others. This is the first 

 condition. 



Next, a curve of this kind will cut the axis CAC 

 in all the points E, G, I, E', G', &c. and only in 

 these. For it can only cut the axis in those points in 

 which y=0, and it will cut it in all these. Besides, 

 when V=0, Qy=z0, and since P Qy=0, therefore 

 P=0, which can only happen when z is one of the 

 roots of the equation P=0 ; that is, as we have al- 

 ready shown, only in the points E, G, I, &c. or 

 E', G', &c. Wherefore the quantity y vanishes, 

 and the curve cuts the axis only in these points. 

 That the curve will cut it in these points, is also clear 

 from this, that in each of them P=0, therefore also 

 Qy=.0, and it is not Q=0, for there is no common 

 root of the equations P=0 and Q=0 ; therefore it 

 must be y=z0, and consequently the curve meets the 

 axis : which fulfils the second condition. 



P 



Besides, since P Qy=z0, y=-fz and if any ab- 

 scissa x be given, z is also given, and therefore P and 

 Q are single and determined, and therefore y is also 

 single and determined. To every absciss x, therefore, 

 there is a corresponding ordinate y, and only one. 

 This is the third condition. 



Again, whether x be assumed positive or negative, 

 while it is of the same length, the value of z-=x 1 is al- 

 ways the same ; and therefore the values of P, Q, and 

 consequently of y, must be the same. So that if equal 

 abscisses be taken on each side of A, the corresponding 

 ordinates will be equal : which is the fourth condition. 



If x be diminished in infinitum, whether it be po- 

 sitive or negative, z will also be diminished in infini- 

 tum, and will be an infinitesimal of the second order. 

 Wherefore in the value of P, all the terms will de- 

 crease in infinitum, f only excepted, because all the 

 rest besides it are multiplied by z ; and thus the va- 

 lue of P will be as yet finite. But the value of Q, 

 which involves the formula drawn entirely into z, will 

 diminish in infinitum, and will become an infinitesimal 



P 



of the second order. Therefore -pzz=u will increase 



.9 

 in infinitum, and becomes an infinitely great quantity 



of the second order. Wherefore the curve will have 

 for its asymptote the straight line AB ; and the area 

 BAED will increase in infinitum, and if the positive 

 ordinates y are taken towards the parts AB, and ex- 

 press the repulsive forces, the asymptotic arc ED 

 will be towards the same parts AB : which was the 

 fifth condition. 



It is clear, then, that however Q be assumed with 

 the given conditions, the first five requisites will be 

 fulfilled. Now the value of Q may be varied in an 

 infinite number of ways, so as still to fulfil th con- 

 ditions with which it was assumed. And therefore 



the arc of the curve intercepted between the in- BoscovichS 

 tersections may be varied in infinite ways, so that }. r J. ot 

 the first five conditions may be fulfilled. It may phjioso- 

 therefore be varied so as to fulfil the sixth condition, nhy. 

 For if there be given however many, and whatever u - ' 

 arches of whatever curves, providing they be such 

 that they recede always from the asymptote AB, 

 and thus no right line parallel to that asymptote cut 

 these arches in more than one point, and in them let 

 there be taken as many points as you please, and as 

 near one another ; it will be easy to assume such a 

 value of P, that the curve may pass through all these 

 points, and the same may be varied infinitely ; so that 

 still the curve will pass through all the same points. 

 Let the number of points assumed be what you please 

 = r, and from every one of such points, let right 

 lines.be drawn parallel to AB, as far as the axis 

 CAC, which must be ordinates of the curve that is 

 sought ; and let the abscisses from A to the said or- 

 dinates be called M 1 , M 2 , M 3 , &c, and the ordinates 

 N 1 , N 1 , N 3 , &c. Let there now be taken a certain 



quantity Az r -f Bs' I +Cz r 2 -(-Gz, and let this 

 quantity be supposed equal to R. Then let another 

 such quantity T be assumed, so that z vanishing, any 

 term of it may vanish, and so that there be no com- 

 mon divisor of the value of P, and of the value of 

 R-f-T, which may be easily done, seeing all the di- 

 visors of the quantity P are known. Let it now be 

 made Q=rR + T, and then the equation of the curve- 

 will be P Ry T=0. After this, let there be 

 put in the equation "m 1 , M 2 , M 3 , successively for x, 

 and N 1 , N 2 , N 3 , &c. for y ; we shall have r equations, 

 each containing values of A, B . C . . . G, of one 

 dimension, besides the given values of M 1 , M 2 , M 3 , 

 &c. N 1 , N 2 , N 3 , &c. and the arbitrary values, which 

 in T are the coefficients of z.- 



By these equations, which are in number r, it will 

 be easy to determine the values of A, B, C, . . . . G, 

 which are likewise in number r, assuming in the first 

 equation, according to the usual method, the value 

 A, and substituting it in all the following equations, 

 by which means the equations will become r 1. 

 These, again, by throwing out the value B, will be 

 reduced to r 2, and so on, until we come to one 

 only, in which the value Q being determined, by 

 means of that, in a retrograde order, all the preceding 

 values will be determined, one by each equation. The 

 values A, B, C, . . . . G being in this manner deter- 

 mined in the equation P R?/ Ty=0, or P Qy=0, 

 it is clear that the values M 1 , M 2 , M 3 , &c. being suc- 

 cessively put for x, the values of the ordinate y must 

 successively be N 1 , N 2 , N 3 , &c. ; and therefore that 

 the curve must pass through these given points in 

 those given curves, and still the value Q will have all 

 the preceding conditions. For z being lessened be- 

 yond whatever limits ; seeing all the terms of the va - 

 lue of T are lessened which were thus assumed, and 

 likewise the terms of the value R are lessened, which 

 are all multiplied by z, and besides this there will be 

 no common divisor of the quantities P and Q, seeing 

 there is none of the quantity P and R-f-P. 



But if two of the nearest of the points assumed in 

 the arches of the curves, on the same side of the axis, 

 be supposed to accede to one another, beyond what- 



