IV. — Ox THE Til. vxscEN DENTAL CiKVEs s'mi/ smmy=asu\XHiunx-\-b. 

 With Plates XIV— XXXVII. By II. A. Newton and A. W. 

 Phillips. 



1. Algebraic curves have been studied hitherto more than trans- 

 ceudeutal. A few of tlie latter have beeu giveu in the text books, 

 but attempts to classify the numerous varieties of transcendental 

 curves have been rare. 



From the form of a transcendental curve it is not easy to state an 

 equation that can represent it. The simpler inverse problem of 

 describing the curve from the equation is naturally the first to be 

 undertaken. The forms that result may, when compared, suggest the 

 solution of the direct problem. We have thought it worth while, 

 therefore, to select for study a single one of the numberless transcen- 

 dental equations, and to exhibit a few of the very many plane curves 

 which that one equation furnishes. 1 he equation selected is, 



sin y sin my =l asinx sin nx-{-b, ( 1 ) 



in which there are four arbitrary constants a, b, m, and n, with two 

 coordinates, x and y. 



2. We assume that ui and u are each less than unity. If either, 



for example >;/, is greater than unity, we may change the unit for y 



in the ratio of 1 :m\)j writing y'=:zmy. The first member of Eq. (l) 



. 1 , . . , , . 1 



then becomes siny' sin — y', where the coefficient of y is — , which is 



less than unity. 



In the equation thus changed, we have assumed in our figures the 

 units for x and y equal, and the axes rectangular. The effect of a 

 different supposition in either particular can be readily understood. 



3. Curves xchose equations <ire y=zasi)ixsinmx. It was found 



convenient to draw several auxiliary curves whose equations are of 



the form, 



y=zaii\\\x^mnix. (2) 



A convenient arlutrary value being assumed for a, to m was given in 

 turn all the values of the proper fractions, which, reduced to their 

 lowest terms, have denominators less than 12. The forms of these 

 curves are shown on plates XIV and XV, excepting a few in which 

 m has 11 for denominator. In fig. 37 is shown the beginning of the 

 curve when m has the irrational value s/\. The axis of x is drawn 

 Trans. Conn. Acad., Vol. III. 13 August, 1875. 



