OF ARTS AND SCIENCES : JANUARY 14, 1873- 499 



would be less than Dx for values of x below a certain fixed value, 

 and x and tan x, starting together from zero, tan x would be less than 

 x for these values, which again is impossible ; 



G= 1. 



Introducing this value of c in [4] [5] and [6] we obtain 



D (sin x) = cos x Dx. {.f~\ 



D (cos x) = — sin x D x. \_g~\ 



D (tan x) = sec 2 x Dx. [A] 



The President communicated the following letter : — 



Cordoba, November 7, 1872. 

 To the President of the American Academy of Arts and 

 Sciences. 



Dear Sir, — Upon my departure from home on the undertaking in 

 which I am at present engaged, the Academy had the goodness to aid 

 my plans materially by appropriating the sum of $ 500 in gold from the 

 Rumford Fund for the purpose of supplying me with a star spectroscope 

 and astronomical photometer, — with permission to transfer them at 

 the same price to the Argentine government for permanent use in this 

 hemisphere, in case that the means for their purchase by the National 

 Observatory should become available. 



The instruments were ordered without delay, and are now in my 

 possession, in good order. But the numerous delays arising from the 

 German war of 1870 postponed both the construction and the transpor- 

 tation of the apparatus ; and the instruments did not reach Cordoba 

 until all my energies had been so severely tasked in other directions as 

 to preclude me from undertaking any spectroscopic or photometric 

 observations for the present. 



I had hoped to offer to the Academy some results obtained with 

 these instruments, in recognition of the valuable aid so opportunely 

 and ^generously afforded ; and I do not yet relinquish the hope of so 

 doing at some future time. Meanwhile I have the satisfaction of being 

 able to announce that the funds have been provided for the acqui- 

 sition of these instruments by the observatory, and have the honor of 

 transmitting them herewith to the Academy, with my cordial thanks. 



It may have some interest for the Academy to know that, in the two 

 years which have elapsed since my arrival in Cordoba, the observa- 



