454 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



be the same, of which those of all the former will be alternately positive and 

 negative after the first term, but those of the latter all negative after the first 

 term; by which means it happens that all the even terms of the former, being 

 positive, will be cancelled by all the equal or corresponding even terms of the 

 latter, which are negative; and hence the sum of the two series, or the value of 

 X, will be denoted by double the sum of the odd terms of either series only. 

 Thus, by the binomial theorem, 



(a + 0^ = «' X (' + 36 - 9ii + 8,i' - 2i5i-^ ^*^-) 



(a - c). = a. X (1 - 3-^ - ^ - g~^7 - .353^, &c.) 



then the sum of these two give, for the value of x, the series 2^a X 



^ 9A« 2436* 65616* ■' 



Now this series is a general expression for the value of x, or the root of the 



equation x^ — qx ■=■ r, in all cases whatever, when a = J-r, and b = -^q, or 



c'^ = a- — b^ = 4-r'^ — -jV?^- When ^r'^ is greater than -bV9^> then c^ or -i-r^ — 



Vt9^ will be positive, and the signs of all the terms of the above series, after the 



first term, will be negative, or to be subtracted from the first term. But when 



4-r'^ is less than Vt9% then c'^ or i-r^ — ^l^^ ^,j]| ^^ negative, and all the even 



terms of the series, will have their signs changed, from minus to plus; by which 



means the series will have its signs alternately positive and negative, after the 



first term, leaving only the 3d, 5th, 7th, &c. terms negative. 



XLIIL Account of the Advantages of a nexvly-invenled Machine much varied 

 in its Ejects, and very Useful for determining the Perfect Proportion between 

 different Moveables acting by Levers and Wheel and Pinion. By Mr. Le Cerf 

 Watch-maker at Geneva, p. QSO. 



This machine may be called a compass of geometrical and mechanical propor- 

 tion; its property is to resolve a great number of problems analogous to the 

 theory and practice of watch-making, in a manner which is at the same time 

 very evident and much more convenient than the mode of arithmetical calcula- 

 tion. It not only shows the proportion of the diameters between the wheels and 

 the pinions, but serves to determine every species of proportion of the calibers, 

 of the size of pivots, width of pallets, magnitude of cylinders, and in general 

 of whatever is an object of dimension. In every case it aftbrds a product, either 

 in unity or fractions, as perfect as the application of it to the mechanism we are 

 considering is easy. 



Nothing therefore can more conduce to the perfection of watch-niaking than 

 an instrument which innnediately determines all the dimensions and proportions 

 required; dimensions and proportions which could not heretofore be obtained 

 but by means so long, so laborious, and so imperfect, that they have probably 



