VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS. 20 



in the series, that constitute any required area, are the same as the conditions 

 of quadribility, and the number of the terms, which agree with the figures 

 whose curves are defined by complete equations. 



Carol. Besides these two series in ^ 2 and 4, for figures of 4 terms, in the 

 same manner may be computed infinite other series, for other cases of figures 

 of 4 terms. Which is also to be understood of all other figures, whose curves 

 are defined by equations consisting of any number of terms. 



There is not time now to give a minute description of the method by which 

 I obtain these series ; yet it may not perhaps be amiss to give some short 

 account of it. Thus, I assume a series composed alike of z and y, viz. 

 Azy + Bzv yi + cz* 2/* + D2^3/*H- &c. = flu. of zy : of which all the terms 

 except the first have general exponents. Then I form an equation between two 

 values of the quantity i, one of which is derived from this series, and the other 

 is easily found, by the direct method of fluxions, from the equation expressing 

 the relation between z and y. From the terms of this equation, properly re- 

 duced, I determine the unknown exponents, p, y, g, h, /, k, &c ; and then the 

 co-efficients a, b, c, &c. And if there be more comparisons than what are 

 sufficient for determining these co-efficients, then from the rest I determine 

 the conditions of quadribility. By proceeding in the right way, the calculation 

 will be very easy. I have also many other rules relating to this matter, which 

 may perhaps appear another time ; as also the use of this method in finding 

 finite irrational quadratures, when rational ones cannot be obtained. 



jibstract of Letters sent to Sir C. H. relating to some Microscopical Observations. 

 Communicated by Sir C. H. to the Editor. N° 284, p. 1357. 



I have made proof of my new set of microscopes, made by Mr. Wilson, 

 and have found the way of applying them very readily. The contrivance of the 

 ivory box and hollow screw for approach, with the illuminating convex at the 

 end of it, is of great service both by day and candle-light ; also the sliders 

 with the plain and concave glass plates for objects are very convenient. But 

 the brass tool wants improvement, viz. a fine threaded steel screw for a more 

 steady approach, and some new turns and motions to the arm which carries the 

 object, &c. also a brass arm to slide up and down on the square rod of the deep 

 microscope, to which I would fix with a screw either the ivory box, or the 

 handle of the brass tool, to be set by that means in a fixed position, to any 

 height or inclination required, which will be very necessary when an illumi- 

 nating glass is applied to either. 



As for the glasses themselves, they are very good, and well wrought, and 



