XVI PREFACE. 



this book. I must, however, mention specially the 

 name of my friend Professor Felix Klein, of Munich, 

 whose private letters have afforded me much valua- 

 ble information, in addition to that derived from his 

 instructive memoirs in the pages of the Mathema- 

 tische Annalen. 



To my friends, Rev. ProfessorTownsend, F. R. S., 

 Professor Everett, Dr. Tarleton, F. T. C. D., Pro- 

 fessor Niven, and Professor Casey, F. R. S., my 

 thanks are due for many useful suggestions and a 

 little friendly criticism, while to the two first-named 

 I am further indebted for their trouble in looking 

 over the proof-sheets in their passage through the 

 press. 



My grateful thanks are also due to the Board of 

 Trinity College for a liberal contribution towards 

 the expenses of publication ; and to the Council of 

 the Royal Irish Academy for kind permission to use 

 the wood engravings which illustrated the original 

 Papers published in their Transactions. 



ROBERT S. BALL. 



THE OBSERVATORY, DUNSINK, Co. DUBLIN, 

 3 1 st December, 1875. 



