VOL. LIII.] PHILOSOPHICAL TRANSACTIONS. 15 



and we take in only a proper number of the first terras of the foregoing series ; 

 but the whole series can never properly express any quantity at all ; because after 

 the 5th term the coefficients begin to increase, and they afterwards increase at a 

 greater rate than what can be compensated by the increase of the powers of z 

 though z represent a number ever so large ; as will be evident by considering the 

 following manner in which the coefficients of that series may be formed. Take 

 a = -^; 5 b = a^ ; 7 = 2 b a; Q d = 2 c a + b*; lle = 2da + 2cb 13/= 

 2ea-\-2db-\-c^; 15g = 2/a + '2 e b-{-2dc, and so on : then take a = a, b = 

 2/;, C=2X3X4c;D = 2X3X4X5X6c?;e = 2X3X4X5x6x7X8(;, 

 and so on: thenA, b, c, d, e, f, &c. will be the coefficients of the foregoing series: 

 whence it easily follows, that if any term in the series after the first 3 be called y, 

 and its distance from the first term n, the next term immediately following will be 



greater than "- ^^ „ " 7 X j^. Therefore at length the subsequent terms of 

 this series are greater than the preceding ones, and increase in infinitum, and 

 therefore the whole series can have no ultimate value whatever. 



Much less can that series have any ultimate value, which is deduced from it 

 by taking z = i, and is supposed to be equal to the logarithm of the square root 

 of the periphery of a circle whose radius is unity ; and what is said concerning the 

 foregoing series is true, and appears to be so, much in the same manner, concerning 

 the series for finding the sum of the logarithms of the odd numbers 3, 5, 7, &c. 

 . . z, and those that are given for finding the sum of the infinite progressions, in 

 which the several terms have the same numerator, while their denominators are 

 any certain power of numbers increasing in arithmetical proportion. But it is 

 needless particularly to insist upon these, because one instance is sufficient to show 

 that those methods are not to be depended on, fi-om which a conclusion follows 

 that is not exact. 



XLir. Of the Insect called the Fegetable Fly* By William Watson, M. D., 



F.R.S. p. 271. 



The beginning of Oct. 1763, Dr. W. received a letter from Dr. Huxham of 

 Plymouth, in which, among other things, he informed him that he lately had ob- 

 tained a sight of what is called the vegetable fly, with the following description 

 of it ; both of which he had from Mr. Newman, an officer of General Duroure's 

 regiment, who came from the island Dominica. As this description seemed to 

 the doctor exceedingly curious, he sent it to Dr. W., exactly transcribed from Mr. 

 Newman's account, which is as follows. 



" The vegetable fly is found in the island Dominica, and (excepting that it has 

 no wings) resembles the drone both in size and colour more than any other En- 



• Fungi, not only of the genus Clavaria, but several others occasionally spring from the bodies of 

 ieai insect*, the seed of such Fungi having accidentally fallen on the insects. 



