6lO PHILOSOPHICAL TRANSACTIONS. [aNNO I/IO. 



subtraction, multiplication, division, or extraction of roots. 2d. By means of 

 the equation so assumed at pleasure, exterminate a out of the general canon, 

 and there will result an equation expressing the relation between the indetermi- 

 nate quantities x and y, 3d. By the known methods take the differential or 

 fluxion of this equation ; and then find the integral or fluent in an infinite 

 series ; which will give the value of x the log. sought. 



Exam. 1. — Assume a =: y \ then, by the general canon, .r = /. (i + y) ; 

 the fluxion of which is .r = j— — ; and the fluent of this in an infinite series 

 \sx=^y - ^y'' -{■ ^y' - -ky' + iy' - i/ + -f/ - &c. 



Exam. 2. — Assume y = ^ ^ ^ ; hence a + I = — —^ ; then, by the gene- 

 ral canon, x = I. , the fluxion of which is i = — ^ — ; and the fluent 



1 — y *^ — yy 



of this expressed in a series is j? = 2 X (y -\- ^y^ + -yy^ -f \y'' + ^y^ -\- &c. 



More examples of this are unnecessary ; since from these it appears how in- 

 numerable logarithmic series may be found, which, without regard to the 

 logarithms of any other numbers, exhibit the log. of the number proposed. 



Lemma 1. — Let z be the log. of any fraction , and x the log. of the 



denominator a -\- \\ then will x = Lb -^ z. Or if z be the log. of the fraction 

 ^ "t , then will x = l.b + z. 



Lemma 2. — Let e be the exponent of any power of the number /?, then will 

 l.b* = e X I'b: therefore, having given the log. of the number b', and the ex- 

 ponent p, the log of the number b will also be given. And both these lemmas 

 appear from the nature of logarithms. 



Part IL — Let a -f- ] be the number, as before, whose log. is to be found ; 

 and let b* be a number produced by the multiplication of numbers, the greatest 

 of which is less than a -\- 1; also let z be the log. of the fraction , that 



is, z = /. ; which equation call the general canon. Then, 1st. for b 



take a quantity any how composed of x and any determinate numbers; and let 

 this value of the number b, so taken at pleasure, be substituted in the 



fraction , whence it will be expressed by a and given numbers. 2d, Let 



there be taken at pleasure any equation between y and a with given numbers ; 

 and by means of this equation exterminate a out of the general canon ; whence 

 will be obtained an equation expressing the relation between the indeterminates 

 z, y. 3d, By the known methods find the differential or fluxion of this equa- 

 tion, and then the integrals or fluents in an infinite series, which will give the 



log. z of the fraction : then, from z, being found, the log. x = Lb — z. 



