Fr T.J. Linane, From Abel to Zundolovich, Vol. 2: Biographies of Priests on the Australian Scene up to 1900, 1980, p 76:

Conway, Cornelius, Irish
Remembered as an Eloquent Preacher
If the ordination date recorded is correct, August 1861, Cornelius must have had a long experience on the Irish Mission before coming to Australia. The first time he appears on the Sydney scene is at the opening of St. Patrick's College, Manly, on 23 January, 1889. Details of origin, education, place of service between 1889 (or before) and 1891 have not come to hand. From 1891-3 he was P.P. of Kogarah, and as from February 1893 to March 1896 he was stationed at St. Francis', Haymarket. It was here that he enjoyed a high reputation as an orator and preacher. After a further stay in Concord (Burwood) from 1896 to 1898 he returned to Ireland. And there the trail ends .. The last detail was picked up from an address of Cardinal James Freeman, given at Concord's centenary celebrations and published in the Catholic Weekly of 23.4.1970. This item has reminded the writer to consult a very well written book on Concord-Burwood, written for the 1970 Centenary by Fr. W.J. Delaney and John Dawes: They Passed This Way. In it we find a tribute to Fr. Conway's zeal and his lively interest in youth. A quote from the Munster News and Limerick Advocate of 30 September 1898 (and reproduced in the above book) takes us a little further along the trail ... "On Friday at St. Patrick's Parish Church. Induction of C. Conway to the joint parishes of St. Patrick's and Maraleen ... Fr. Conway is a member of an old Limerick family and has recently returned from Sydney ...."
(Footprints, Vol. 2, No. 10)
Ref.: As Quoted.