180 PHILOSOPHICAL TRANSACTIONS. [aNNO 16Q7 . 



any product whereof any power of the infinite series az + bzz + cz^ &c. is 

 composed, the number which expresses how many ways the letters of each pro- 

 duct may be changed. 



Now to find how many ways the letters of any product, for instance 

 a"» - "b''cPd'', may be changed ; this is the rule which is commonly given : write 

 as many terms of the series 1 X2X3X4X5 &c. as there are units in the 

 sum of the indices, viz. m — 7i -\- h -\- p -\- r ; let this series be the numera- 

 tor of a fraction, whose denominator shall be the product of the series 1 X 2 

 X3X4&C. 1X2X3X4X5 &c. 1X2 X 3X4X5X6 &c. 

 whereof the first is to contain as many terms, as there are units in the first in- 

 dex m — n; the second as many as there are units in the second index k; the 

 third as many as there are units in the third index p ; the fourth as many as 

 there are units in the 4th index r. But the numerator and denominator of this 

 fraction have a common divisor, viz. the series 1X2X3X4X5 &c. con- 

 tinued to so many terms as there are units in the first index m — 7i; therefore 

 let both this numerator and denominator be divided by this common divisor, 

 then this new numerator will begin with m — n -f- 'j whereas the other began 

 with 1, and will contain so many terms as there are units in h -\- p -f- r, that 

 is as many as there are units in the sum of all the indices, excepting the first. 

 As for the new denominator, it will be the product of 3 series only, that is of 

 so many as there are indices, excepting the first. But if it happen that n is equal 

 to A -j- /) -|- r, as it always happens in our theorem, then the numerator be- 

 ginning by m — n -\- I, and being continued to as many terms as there are 

 units \n h -\- p -\- r or n, the last term will necessarily be m ; so if you invert 

 the series, and make that the first term which was the last, the numerator will 

 he. m y. m — 1 y. m — 1 X m — 3, &c. continued to so many terms as there 

 are units in the sum of the indices of each product, excepting the first index. 

 There remains but one thing to demonstrate, which is, that what I have said of 

 powers whose index is an integer, may be adapted to roots, or powers whose 

 index is a fraction ; but it appears at first sight why it should be so : for the 

 same reason which makes me consider roots under the notion of powers, will 

 make me conclude, that whatever is said of one may be said of the other. 

 However I think to give some time a more formal demonstration of it. 



Of an Error committed by common Surveyors, in comparing of Surveys taken at 

 long Intervals of Time, arising from the Fariation of the Magnetic Needle. 

 By William Molyneux, Esq. F. R. S. N° 230, p. 625. 

 The variation of the magnetic needle is so commonly known, that I need not 



insist much on its explication. It is certain that the true solar meridian, and 



