206 Mr. A. R. Forsyth. On the Theta- Functions, [Dec. 22 r 



Substance. Formula. M. P. m. 



Dichlorobenzol .... C 6 H 4 C1 2 =147 - . 52-821 . . -3593 3 



Bromaniline C 6 H 6 BrN=172 .. 61-742 .. -35971 



Trinitrotoluol C 7 H 3 N S 6 =227 .. 80-532 .. -35477 



Here the first pair of values of m are almost identical. It is evident,, 

 however, that this simple relation does not generally prevail ; indeed, 

 in the case of isomeric substances, melting point may alter widely, 

 while additive formula remains constant. 



The following are examples of the identification of series by melting 



point : — 



M. P. M. P. 



*- Trinitrotoluol 78'853- *-Dinitrotoluol 69-252=9*601 "1 



Trinifcrophenol 121*194— 0-Dinitrophenol 111-621=9-573 J 



*-Dinitrotoluol 69*252— Nitrotoluol 51-407 = 17*8451 



^-Dinitrophenol 61*843— a-Nitrophenol 44-392=17*451 J 



The melting points recorded in the memoir are important physical 

 constants, now first determined with a small probable error, and with 

 an apparatus of considerable simplicity. Under no range of ordinary 

 atmospheric pressure or latitude, and in no ordinary interval of time,, 

 are these constants likely to become impaired. Hence, if the sub- 

 stances referred to be prepared and preserved with average care, and 

 handled with moderate skill, they constitute in themselves a set of 

 thermometric standards, distributed at mean intervals of about 4° 

 between 42° and 120°. If these substances, or most of them, be at 

 hand, they enable an investigator to at once calibrate and directly 

 refer to the air thermometer any standard mercurial instrument, without 

 the necessary application of any correction whatever. 



IV. " Memoir on the Theta-Functions, particularly those of Two 

 Variables." By A. R. Forsyth, B.A., Fellow of Trinity 

 College, Cambridge. Communicated by A. Cayley, LL.D. y 

 F.R.S. Received December 9, 1881. 



(Abstract.) 



The paper of which this is an abstract is divided into four parts, to 

 the whole being prefixed a list of the more important papers dealing 

 with the double theta-functions. 



Section I treats of what may be called Rosenhain's theory, and its- 

 object is to obtain from a more general basis, and in an easier manner, 

 the results given by Rosenhain in his essay " Memoire sur les Fonc- 

 tions des Deux Variables et a Quatre Periodes," which obtained the 



